Difference between revisions of "Cosets and Lagrange’s Theorem"

Latest revision as of 15:34, 17 November 2021

Left and Right Cosets of Subgroups

Definition: Let $(G,\cdot )$ be a group and let $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (H, \cdot)}$ be a subgroup. Let $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g \in G}$ . Then the Left Coset of $H$ with Representative $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g}$ is the set $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle gH = \{ gh : h \in H \}}$ . The Right Coset of $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H}$ with Representative $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g}$ is the set $Hg=\{hg:h\in H\}$ .

When the operation symbol “ $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle +}$ ” is used instead of $\cdot$ we often denote the left and right cosets of $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H}$ with representation $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g}$ with the notation $g+H$ and $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H + g}$ respectively.

For example, consider the group $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\mathbb{Z}, +)}$ and the subgroup $(3\mathbb {Z} ,+)$ . Consider the element $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2 \in \mathbb{Z}}$ . Then the left coset of $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3\mathbb{Z}}$ with representative $2$ is:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad 2 + 3\mathbb{Z} = \{2 + h : h \in 3 \mathbb{Z} \} = \{ ..., -1, 2, 5, ... \} \end{align}}

And the right coset of $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3 \mathbb{Z}}$ with representative $2$ is:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad 3\mathbb{Z} + 2 = \{ h + 2 : h \in 3 \mathbb{Z} \} = \{ ..., -1, 2, 5, ... \} \end{align}}

In this particular example we see that $2+3\mathbb {Z} =3\mathbb {Z} +2$ . But in general, is $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle gH = Hg}$ for a given subgroup $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (H, \cdot)}$ of $(G,\cdot )$ and for $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g \in G}$ ? The answer is NO. There are many examples when left cosets are not equal to corresponding right cosets.

To illustrate this, consider the symmetric group $(S_{3},\circ )$ . Let $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G = \{ \epsilon, (12) \}}$ . Then $(G,\circ )$ is a subgroup of $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (S_3, \circ)}$ since $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G \subset S_3}$ , $G$ is closed under $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \circ}$ , and $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon^{-1} = \epsilon}$ , $(12)^{-1}=(12)$ (since $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (12)}$ is a transposition). Now consider the element $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (13) \in S_3}$ . Then the left coset of $G$ with representative $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (13)}$ is:

{\begin{aligned}\quad (13)G=\{(13)\circ h:h\in G\}=\{(13)\circ \epsilon ,(13)\circ (12)\}=\{(13),(123)\}\end{aligned}}

And the right coset of $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G}$ with representative $(13)$ is:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad G(13) = \{ h \circ (13) : h \in G \} = \{ \epsilon \circ (13), (12) \circ (13) \} = \{ (13), (132) \} \end{align}}

We note that $(123)\neq (132)$ and so $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (13)G \neq G(13)}$ !

So, when exactly are the left and right cosets of a subgroup with representative $g$ equal? The following theorem gives us a simple criterion for a large class of groups.

Proposition 1: Let $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (G, \cdot)}$ be a group and let $(H,\cdot )$ be a subgroup. If $(G,\cdot )$ is abelian then for all $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g \in G}$ , $gH=Hg$ .

Proof: Let $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g \in G}$ . If $G$ is abelian then for all $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h \in G}$ (and hence for all $h\in H$ ) we have that $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g \cdot h = h \cdot g}$ . So:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad gH = \{ g \cdot h : h \in H \} = \{ h \cdot g : h \in H \} = Hg \quad \blacksquare \end{align}}

Proposition 2: Let $(G,\cdot )$ be a group, $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (H, \cdot)}$ a subgroup, and $g_{1},g_{2}\in G$ . Then the following statements are equivalent:

a) $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g_1H = g_2H}$ .
b) $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Hg_1^{-1} = Hg_2^{-1}}$ .
c) $g_{1}H\subseteq g_{2}H$ .
d) $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g_1 \in g_2H}$ .

e) $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g_1^{-1}g_2 \in H}$ .

The Set of Left (Right) Cosets of a Subgroup Partitions the Whole Group

Recall that if $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (G, \cdot)}$ is a group, $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (H, \cdot)}$ is a subgroup, and $g\in G$ then the left coset of $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H}$ with representative $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g}$ is the set:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad gH = \{ gh : h \in H \} \end{align}}

The right coset of $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H}$ with representative $g$ is the set:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad Hg = \{ hg : h \in H \} \end{align}}

We will now look at a nice theorem which tells us that the set of all left cosets of a subgroup $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (H, \cdot)}$ actually partitions $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (G, \cdot)}$ . The proof below can be mirrored to analogously show that the set of all right cosets of a subgroup $(H,\cdot )$ also partitions $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (G, \cdot)}$ .

Theorem 1: Let $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (G, \cdot)}$ be a group and let $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (H, \cdot)}$ a subgroup. Then the set of all left cosets of $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (H, \cdot)}$ partitions $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (G, \cdot)}$ .

Recall that a partition of a set $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A}$ is a collection of nonempty subsets of $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A}$ that are pairwise disjoint and whose union is all of $A$ .

Proof: We first show that any two distinct left cosets of $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H}$ are disjoint. Let $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g_1, g_2 \in G}$ , $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g_1 \neq g_2}$ and assume the left cosets $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g_1H}$ and $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g_2H}$ are distinct. Suppose that $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g_1H \cap g_2H \neq \emptyset}$ . Then there exists an $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in g_1H \cap g_2H}$ . So $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in g_1H}$ and $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in g_2H}$ and there exists $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h_1, h_2 \in H}$ such that:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad x = g_1h_1 \quad (*) \end{align}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad x = g_2h_2 \quad (**) \end{align}}

So using $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (*)}$ and $(**)$ we see that $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g_1h_1 = g_2h_2}$ . So $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g_1 = g_2h_2h_1^{-1}}$ . But $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h_2h_1^{-1} \in H}$ since $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (H, \cdot)}$ is a group and is hence closed under $\cdot$ . So $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g_1 \in g_2H}$ . But this means that $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g_1H = g_2H}$ , a contradiction since $g_{1}H$ and $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g_2H}$ distinct. So the assumption that $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g_1H \cap g_2H \neq \emptyset}$ was false. So:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad g_1H \cap g_2H = \emptyset \end{align}}

Now if $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i}$ is the identity element for $(G,\cdot )$ then $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i \in H}$ since $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (H, \cdot)}$ is a subgroup and must contain the identity element. So $g=gi\in gH$ for all $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g \in G}$ . So:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad G = \bigcup_{g \in G} gH \end{align}}

Therefore the left cosets of $(H,\cdot )$ partition $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (G, \cdot)}$ .

The Number of Elements in a Left (Right) Coset

Recall that if $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (G, \cdot)}$ is a group, $(H,\cdot )$ is a subgroup, and $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g \in G}$ then the left coset of $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H}$ with representative $g$ is defined as:

{\begin{aligned}\quad gH=\{gh:h\in H\}\end{aligned}}

The right coset of $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H}$ with representative $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g}$ is defined as:

{\begin{aligned}\quad Hg=\{hg:h\in H\}\end{aligned}}

We will now look at a rather simple theorem which will tell us that the number of elements in a left (or right coset) will equal to the number of elements in $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H}$ . This seems rather obvious since $gH$ contains elements of the form $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle gh}$ where we range through all of $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h}$ . So $gH$ has at most the same number of elements in $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H}$ . Of course, $gH$ may have less elements if $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle gh_1 = gh_2}$ for distinct $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h_1, h_2 \in H}$ . Of course this cannot be since by cancellation we would then have that $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h_1 = h_2}$ . We make this argument more rigorous below.

Theorem 1: Let $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (G, \cdot)}$ be a group, $(H,\cdot )$ a subgroup, and let $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g \in G}$ . Then the number of elements in $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle gH}$ equals the number of elements in $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H}$ , i.e., $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mid gH \mid = \mid H \mid}$ . Similarly, $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mid Hg \mid = \mid H \mid}$ .

We only prove the case of this theorem for left cosets. The case for right cosets is analogous.

Proof: Define a function $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f : H \to gH}$ for all $h\in H$ by:

{\begin{aligned}\quad f(h)=gh\end{aligned}}

We will show that $f$ is bijective. First we show that $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f}$ is injective. Let $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h_1, h_2 \in H}$ and assume that $f(h_{1})=f(h_{2})$ . Then we have that:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad gh_1 = gh_2 \end{align}}

By left cancellation this implies that $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h_1 = h_2}$ and so $f$ is injective. We now show that $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f}$ is surjective. Let $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle gh \in gH}$ . Then $f(h)=gh$ (somewhat trivially) so $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f}$ is surjective.

Since $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f}$ is bijective we have that $\mid H\mid =\mid gH\mid$ so the number of elements in the left coset $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle gH}$ is equal to the number of elements in $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H}$ . $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \blacksquare}$

The Index of a Subgroup

If $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (G, \cdot)}$ is a group and $(H,\cdot )$ is a subgroup then we might want to know the number of left cosets and the number of right cosets of $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H}$ . As the following theorem will show - the number of left cosets of $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H}$ will always equal the number of right cosets of $H$ .

Theorem 1: Let $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (G, \cdot)}$ be a group and let $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (H, \cdot)}$ be a subgroup. Then the number of left cosets of $H$ equals the number of right cosets of $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H}$ .

Proof: Let $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L_H}$ denote the set of all left cosets of $H$ and let $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R_H}$ denote the set of all right cosets of $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H}$ . Define a function $f:L_{H}\to R_{H}$ for all $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle gH \in L_H}$ by

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad f(gH) = Hg^{-1} \end{align}}

If we can show that $f$ is bijective then $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mid L_H \mid = \mid R_H}$ . We first show that $f$ is injective. Let $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g_1H, g_2H \in L_H}$ and suppose that $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(g_1H) = f(g_2H)}$ . Then:

{\begin{aligned}\quad Hg_{1}^{-1}=Hg_{2}^{-1}\end{aligned}}

But then by the theorem of equivalent statements presented on the <a href="/left-and-right-cosets-of-subgroups">Left and Right Cosets of Subgroups</a> page we must have that:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad g_1H = g_2H \end{align}}

Hence $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f}$ is injective. We now show that $f$ is surjective. Let $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Hg \in R_H}$ . Then we have that for $g^{-1}H\in L_{H}$ that:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad f(g^{-1}H) = H(g^{-1})^{-1} = Hg \end{align}}

So $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f}$ is indeed surjective. Since $f$ is a function from a finite set to a finite set that is bijective we must have that $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mid L_H \mid = \mid R_H \mid}$ , i.e., the number of left cosets of $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H}$ equals the number of right cosets of $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H}$ . $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \blacksquare}$

With the result above, we can unambiguously define the index of subgroup $(H,\cdot )$ in a group $(G,\cdot )$ .

Definition: Let $(G,\cdot )$ be a group and let $(H,\cdot )$ be a subgroup. The Index of $H$ in $G$ denoted $[G:H]$ is defined as the number of left (or right) cosets of $H$ .

For example, consider the symmetric group $(S_{3},\circ )$ and let $H=\{\epsilon ,(12)\}$ . Let's find the index $S_{3}:H]$ . Note that $S_{3}=\{\epsilon ,(12),(13),(23),(123),(132)\}$ . So the left cosets of $H$ in $S_{3}$ are:

{\begin{aligned}\quad \epsilon H=\{\epsilon \circ \epsilon ,\epsilon \circ (12)\}=\{\epsilon ,(12)\}\\\end{aligned}}

{\begin{aligned}\quad (12)H=\{(12)\circ \epsilon ,(12)\circ (12)\}=\{(12),\epsilon \}\end{aligned}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad (13)H = \{ (13) \circ \epsilon, (13) \circ (12) \} = \{ (13), (123) \} \end{align}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad (23)H = \{ (23) \circ \epsilon, (23) \circ (12) \} = \{ (23), (132) \} \end{align}}

{\begin{aligned}\quad (123)H=\{(123)\circ \epsilon ,(123)\circ (12)\}=\{(123),(13)\}\end{aligned}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad (132)H = \{ (132) \circ \epsilon, (132) \circ (12) \} = \{ (132), (23) \} \end{align}}

So the set of left cosets of $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H}$ is:

{\begin{aligned}\quad \{\{\epsilon ,(12)\},\{(13),(123)\},\{(23),(132)\}\}\end{aligned}}

There are three such cosets so $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [S_3 : H] = 3}$ .

For another example, consider the group $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\mathbb{Z}, +)}$ and the subgroup $(3\mathbb {Z} ,+)$ . Then the left cosets of $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (3\mathbb{Z}, +)}$ are:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad 0 + 3\mathbb{Z} = \{ ..., -3, 0, 3, ... \} \end{align}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad 1 + 3\mathbb{Z} = \{ ..., -2, 1, 4, ... \} \end{align}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad 2 + 3\mathbb{Z} = \{ ..., -1, 2, 5, ... \} \end{align}}

Note that if $m\equiv n{\pmod {3}}$ then $m+3\mathbb {Z} =n+3\mathbb {Z}$ . So in this example we have that $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [\mathbb{Z} : 3\mathbb{Z}] = 3}$ .

Lagrange's Theorem

We now have all of the tools to prove a very important and astonishing theorem regarding subgroups. This theorem is known as Lagrange's theorem and will tell us that the number of elements in a subgroup of a larger group must divide the number of elements in the larger group.

Theorem 1 (Lagrange's Theorem): Let $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (G, \cdot)}$ be a finite group and let $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (H, \cdot)}$ be a subgroup. Then the number of elements in $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H}$ must divide the number of elements in $G$ .

Proof:The set of left cosets of $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H}$ partition $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G}$ , that is for all $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g_1, g_2 \in G}$ with $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g_1 \neq g_2}$ we have that, $g_{1}H\cap g_{2}H=\emptyset$ and:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad G = \bigcup_{g \in G} gH \end{align}}

The number of left cosets of $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H}$ is the index $[G:H]$ and the number of elements in $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle gH}$ is equal to the number of elements in $H$ . So:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \quad \mid G \mid = [G : H] \mid H \mid \end{align}}

Therefore $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mid H \mid}$ divides $\mid G\mid$ , i.e., the number of elements in any subgroup $(H,\cdot )$ of a finite group $(G,\cdot )$ must divide the number of elements in $(G,\cdot )$ . $\blacksquare$

Corollaries to Lagrange's Theorem

Recall that if $(G,\cdot )$ is a finite group and $(H,\cdot )$ is a subgroup then the number of elements in $H$ must divide the number of elements in $G$ and moreover:

{\begin{aligned}\quad \mid G\mid =[G:H]\mid H\mid \end{aligned}}

We will now prove some amazing corollaries relating to Lagrange's theorem.

Corollary 1: Let $(G,\cdot )$ be a finite group and let $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (H, \cdot)}$ be a subgroup. Then the index of $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H}$ is $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G}$ is given by $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \displaystyle{[G : H] = \frac{\mid G \mid}{\mid H \mid}}}$ .

Proof: Rearrange the formula in the proof of Lagrange's theorem. $\blacksquare$

Corollary 2: Let $(G,\cdot )$ be a finite group and let $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g \in G}$ . Then the order of the element $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g}$ must divide $\mid G\mid$ .

The order of an element $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g}$ is the smallest positive integer $n$ such that $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g^n = e}$ (where of course, $e$ is the identity element for the group.

Proof: The order of the element $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g}$ is the smallest positive integer $n$ such that $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g^n = e}$ . The generated subgroup $(<g>,\cdot )$ is such that $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mid <g> \mid = n}$ . So by Lagrange's theorem $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n = \mid < g > \mid}$ must divide $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mid G \mid}$ , i.e., the order of $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g}$ must divide $\mid G\mid$ .

Corollary 3: Let $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (G, \cdot)}$ be a finite group of order $\mid G\mid =p$ where $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p}$ is a prime number. Then $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G}$ is a cyclic group.

Proof: Let $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g \in G}$ be any non-identity element in $G$ (which exists since $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p \geq 2}$ ). By Lagrange's Theorem, the subgroup $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (<g>, \cdot)}$ must be such that $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mid <g> \mid}$ divides $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mid G \mid = p}$ . But the only positive divisors of $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p}$ are $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1}$ and $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p}$ .

So if $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mid <g> \mid = 1}$ then $g=e$ which is a contradiction since we assumed that $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g}$ is not the identity for the group. So $\mid <g>\mid =p$ , i.e., $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G = <g>}$ . So $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G}$ is a cyclic group.

Corollary 4: Let $(G,\cdot )$ be a finite group and let $(H,\cdot )$ and $(I,\cdot )$ be subgroups of $G$ such that $I\subseteq H\subseteq G$ . Then $[G:I]=[G:H][H:I]$ .

Proof: By Lagrange's theorem we have that:

{\begin{aligned}\quad [G:I]&={\frac {\mid G\mid }{\mid I\mid }}={\frac {\mid G\mid }{\mid H\mid }}\cdot {\frac {\mid H\mid }{\mid I\mid }}=[G:H][H:I]\quad \blacksquare \end{aligned}}

Licensing

Content obtained and/or adapted from:

Left and Right Cosets of Subgroups, mathonline.wikidot.com under a CC BY-SA license
The Set of Left (Right) Cosets of a Subgroup Partitions the Whole Group, mathonline.wikidot.com under a CC BY-SA license
The Number of Elements in a Left (Right) Coset, mathonline.wikidot.com under a CC BY-SA license
The Index of a Subgroup, mathonline.wikidot.com under a CC BY-SA license
Lagrange's Theorem, mathonline.wikidot.com under a CC BY-SA license
Corollaries to Lagrange's Theorem, mathonline.wikidot.com under a CC BY-SA license

@@ Line 31: / Line 31: @@
 </blockquote>
-==The Set of Left (Right) Cosets of a Subgroup Partitions the Whole Group==
+===The Set of Left (Right) Cosets of a Subgroup Partitions the Whole Group===
 <p>Recall that if <span class="math-inline"><math>(G, \cdot)</math></span> is a group, <span class="math-inline"><math>(H, \cdot)</math></span> is a subgroup, and <span class="math-inline"><math>g \in G</math></span> then the left coset of <span class="math-inline"><math>H</math></span> with representative <span class="math-inline"><math>g</math></span> is the set:</p>
 <div style="text-align: center;"><math>\begin{align} \quad gH = \{ gh : h \in H \} \end{align}</math></div>
@@ Line 58: / Line 58: @@
 </ul>
+===The Number of Elements in a Left (Right) Coset===
+<p>Recall that if <span class="math-inline"><math>(G, \cdot)</math></span> is a group, <span class="math-inline"><math>(H, \cdot)</math></span> is a subgroup, and <span class="math-inline"><math>g \in G</math></span> then the left coset of <span class="math-inline"><math>H</math></span> with representative <span class="math-inline"><math>g</math></span> is defined as:</p>
+<div style="text-align: center;"><math>\begin{align} \quad gH = \{ gh : h \in H \} \end{align}</math></div>
+<p>The right coset of <span class="math-inline"><math>H</math></span> with representative <span class="math-inline"><math>g</math></span> is defined as:</p>
+<div style="text-align: center;"><math>\begin{align} \quad Hg = \{ hg : h \in H \} \end{align}</math></div>
+<p>We will now look at a rather simple theorem which will tell us that the number of elements in a left (or right coset) will equal to the number of elements in <span class="math-inline"><math>H</math></span>. This seems rather obvious since <span class="math-inline"><math>gH</math></span> contains elements of the form <span class="math-inline"><math>gh</math></span> where we range through all of <span class="math-inline"><math>h</math></span>. So <span class="math-inline"><math>gH</math></span> has at most the same number of elements in <span class="math-inline"><math>H</math></span>. Of course, <span class="math-inline"><math>gH</math></span> may have less elements if <span class="math-inline"><math>gh_1 = gh_2</math></span> for distinct <span class="math-inline"><math>h_1, h_2 \in H</math></span>. Of course this cannot be since by cancellation we would then have that <span class="math-inline"><math>h_1 = h_2</math></span>. We make this argument more rigorous below.</p>
+<blockquote style="background: white; border: 1px solid black; padding: 1em;">
+<td><strong>Theorem 1:</strong> Let <span class="math-inline"><math>(G, \cdot)</math></span> be a group, <span class="math-inline"><math>(H, \cdot)</math></span> a subgroup, and let <span class="math-inline"><math>g \in G</math></span>. Then the number of elements in <span class="math-inline"><math>gH</math></span> equals the number of elements in <span class="math-inline"><math>H</math></span>, i.e., <span class="math-inline"><math>\mid gH \mid = \mid H \mid</math></span>. Similarly, <span class="math-inline"><math>\mid Hg \mid = \mid H \mid</math></span>.</td>
+</blockquote>
+<p><em>We only prove the case of this theorem for left cosets. The case for right cosets is analogous.</em></p>
+<ul>
+<li><strong>Proof:</strong> Define a function <span class="math-inline"><math>f : H \to gH</math></span> for all <span class="math-inline"><math>h \in H</math></span> by:</li>
+</ul>
+<div style="text-align: center;"><math>\begin{align} \quad f(h) = gh \end{align}</math></div>
+<ul>
+<li>We will show that <span class="math-inline"><math>f</math></span> is bijective. First we show that <span class="math-inline"><math>f</math></span> is injective. Let <span class="math-inline"><math>h_1, h_2 \in H</math></span> and assume that <span class="math-inline"><math>f(h_1) = f(h_2)</math></span>. Then we have that:</li>
+</ul>
+<div style="text-align: center;"><math>\begin{align} \quad gh_1 = gh_2 \end{align}</math></div>
+<ul>
+<li>By left cancellation this implies that <span class="math-inline"><math>h_1 = h_2</math></span> and so <span class="math-inline"><math>f</math></span> is injective. We now show that <span class="math-inline"><math>f</math></span> is surjective. Let <span class="math-inline"><math>gh \in gH</math></span>. Then <span class="math-inline"><math>f(h) = gh</math></span> (somewhat trivially) so <span class="math-inline"><math>f</math></span> is surjective.</li>
+</ul>
+<ul>
+<li>Since <span class="math-inline"><math>f</math></span> is bijective we have that <span class="math-inline"><math>\mid H \mid = \mid gH \mid</math></span> so the number of elements in the left coset <span class="math-inline"><math>gH</math></span> is equal to the number of elements in <span class="math-inline"><math>H</math></span>. <span class="math-inline"><math>\blacksquare</math></span></li>
+</ul>
+===The Index of a Subgroup===
+<p>If <span class="math-inline"><math>(G, \cdot)</math></span> is a group and <span class="math-inline"><math>(H, \cdot)</math></span> is a subgroup then we might want to know the number of left cosets and the number of right cosets of <span class="math-inline"><math>H</math></span>. As the following theorem will show - the number of left cosets of <span class="math-inline"><math>H</math></span> will always equal the number of right cosets of <span class="math-inline"><math>H</math></span>.</p>
+<blockquote style="background: white; border: 1px solid black; padding: 1em;">
+<td><strong>Theorem 1:</strong> Let <span class="math-inline"><math>(G, \cdot)</math></span> be a group and let <span class="math-inline"><math>(H, \cdot)</math></span> be a subgroup. Then the number of left cosets of <span class="math-inline"><math>H</math></span> equals the number of right cosets of <span class="math-inline"><math>H</math></span>.</td>
+</blockquote>
+<ul>
+<li><strong>Proof:</strong> Let <span class="math-inline"><math>L_H</math></span> denote the set of all left cosets of <span class="math-inline"><math>H</math></span> and let <span class="math-inline"><math>R_H</math></span> denote the set of all right cosets of <span class="math-inline"><math>H</math></span>. Define a function <span class="math-inline"><math>f : L_H \to R_H</math></span> for all <span class="math-inline"><math>gH \in L_H</math></span> by</li>
+</ul>
+<div style="text-align: center;"><math>\begin{align} \quad f(gH) = Hg^{-1} \end{align}</math></div>
+<ul>
+<li>If we can show that <span class="math-inline"><math>f</math></span> is bijective then <span class="math-inline"><math>\mid L_H \mid = \mid R_H</math></span>. We first show that <span class="math-inline"><math>f</math></span> is injective. Let <span class="math-inline"><math>g_1H, g_2H \in L_H</math></span> and suppose that <span class="math-inline"><math>f(g_1H) = f(g_2H)</math></span>. Then:</li>
+</ul>
+<div style="text-align: center;"><math>\begin{align} \quad Hg_1^{-1} = Hg_2^{-1} \end{align}</math></div>
+<ul>
+<li>But then by the theorem of equivalent statements presented on the <a href="/left-and-right-cosets-of-subgroups">Left and Right Cosets of Subgroups</a> page we must have that:</li>
+</ul>
+<div style="text-align: center;"><math>\begin{align} \quad g_1H = g_2H \end{align}</math></div>
+<ul>
+<li>Hence <span class="math-inline"><math>f</math></span> is injective. We now show that <span class="math-inline"><math>f</math></span> is surjective. Let <span class="math-inline"><math>Hg \in R_H</math></span>. Then we have that for <span class="math-inline"><math>g^{-1}H \in L_H</math></span> that:</li>
+</ul>
+<div style="text-align: center;"><math>\begin{align} \quad f(g^{-1}H) = H(g^{-1})^{-1} = Hg \end{align}</math></div>
+<ul>
+<li>So <span class="math-inline"><math>f</math></span> is indeed surjective. Since <span class="math-inline"><math>f</math></span> is a function from a finite set to a finite set that is bijective we must have that <span class="math-inline"><math>\mid L_H \mid = \mid R_H \mid</math></span>, i.e., the number of left cosets of <span class="math-inline"><math>H</math></span> equals the number of right cosets of <span class="math-inline"><math>H</math></span>. <span class="math-inline"><math>\blacksquare</math></span></li>
+</ul>
+<p>With the result above, we can unambiguously define the index of subgroup <span class="math-inline"><math>(H, \cdot)</math></span> in a group <span class="math-inline"><math>(G, \cdot)</math></span>.</p>
+<blockquote style="background: white; border: 1px solid black; padding: 1em;">
+<td><strong>Definition:</strong> Let <span class="math-inline"><math>(G, \cdot)</math></span> be a group and let <span class="math-inline"><math>(H, \cdot)</math></span> be a subgroup. The <strong>Index of <span class="math-inline"><math>H</math></span> in <span class="math-inline"><math>G</math></span></strong> denoted <span class="math-inline"><math>[G : H]</math></span> is defined as the number of left (or right) cosets of <span class="math-inline"><math>H</math></span>.</td>
+</blockquote>
+<p>For example, consider the symmetric group <span class="math-inline"><math>(S_3, \circ)</math></span> and let <span class="math-inline"><math>H = \{ \epsilon, (12) \}</math></span>. Let's find the index <span class="math-inline"><math>S_3 : H ]</math></span>. Note that <span class="math-inline"><math>S_3 = \{ \epsilon, (12), (13), (23), (123), (132) \}</math></span>. So the left cosets of <span class="math-inline"><math>H</math></span> in <span class="math-inline"><math>S_3</math></span> are:</p>
+<div style="text-align: center;"><math>\begin{align} \quad \epsilon H = \{ \epsilon \circ \epsilon, \epsilon \circ (12) \} = \{ \epsilon, (12) \} \\ \end{align}</math></div>
+<div style="text-align: center;"><math>\begin{align} \quad (12)H= \{ (12) \circ \epsilon, (12) \circ (12) \} = \{ (12), \epsilon \} \end{align}</math></div>
+<div style="text-align: center;"><math>\begin{align} \quad (13)H = \{ (13) \circ \epsilon, (13) \circ (12) \} = \{ (13), (123) \} \end{align}</math></div>
+<div style="text-align: center;"><math>\begin{align} \quad (23)H = \{ (23) \circ \epsilon, (23) \circ (12) \} = \{ (23), (132) \} \end{align}</math></div>
+<div style="text-align: center;"><math>\begin{align} \quad (123)H = \{ (123) \circ \epsilon, (123) \circ (12) \} = \{ (123), (13) \} \end{align}</math></div>
+<div style="text-align: center;"><math>\begin{align} \quad (132)H = \{ (132) \circ \epsilon, (132) \circ (12) \} = \{ (132), (23) \} \end{align}</math></div>
+<p>So the set of left cosets of <span class="math-inline"><math>H</math></span> is:</p>
+<div style="text-align: center;"><math>\begin{align} \quad \{\{ \epsilon, (12) \}, \{ (13), (123) \}, \{ (23), (132) \} \} \end{align}</math></div>
+<p>There are three such cosets so <span class="math-inline"><math>[S_3 : H] = 3</math></span>.</p>
+<p>For another example, consider the group <span class="math-inline"><math>(\mathbb{Z}, +)</math></span> and the subgroup <span class="math-inline"><math>(3\mathbb{Z}, +)</math></span>. Then the left cosets of <span class="math-inline"><math>(3\mathbb{Z}, +)</math></span> are:</p>
+<div style="text-align: center;"><math>\begin{align} \quad 0 + 3\mathbb{Z} = \{ ..., -3, 0, 3, ... \} \end{align}</math></div>
+<div style="text-align: center;"><math>\begin{align} \quad 1 + 3\mathbb{Z} = \{ ..., -2, 1, 4, ... \} \end{align}</math></div>
+<div style="text-align: center;"><math>\begin{align} \quad 2 + 3\mathbb{Z} = \{ ..., -1, 2, 5, ... \} \end{align}</math></div>
+<p>Note that if <span class="math-inline"><math>m \equiv n \pmod 3</math></span> then <span class="math-inline"><math>m + 3\mathbb{Z} = n + 3\mathbb{Z}</math></span>. So in this example we have that <span class="math-inline"><math>[\mathbb{Z} : 3\mathbb{Z}] = 3</math></span>.</p>
+==Lagrange's Theorem==
+<p>We now have all of the tools to prove a very important and astonishing theorem regarding subgroups. This theorem is known as Lagrange's theorem and will tell us that the number of elements in a subgroup of a larger group must divide the number of elements in the larger group.</p>
+<blockquote style="background: white; border: 1px solid black; padding: 1em;">
+<td><strong>Theorem 1 (Lagrange's Theorem):</strong> Let <span class="math-inline"><math>(G, \cdot)</math></span> be a finite group and let <span class="math-inline"><math>(H, \cdot)</math></span> be a subgroup. Then the number of elements in <span class="math-inline"><math>H</math></span> must divide the number of elements in <span class="math-inline"><math>G</math></span>.</td>
+</blockquote>
+<ul>
+<li><strong>Proof:</strong>The set of left cosets of <span class="math-inline"><math>H</math></span> partition <span class="math-inline"><math>G</math></span>, that is for all <span class="math-inline"><math>g_1, g_2 \in G</math></span> with <span class="math-inline"><math>g_1 \neq g_2</math></span> we have that, <span class="math-inline"><math>g_1H \cap g_2H = \emptyset</math></span> and:</li>
+</ul>
+<div style="text-align: center;"><math>\begin{align} \quad G = \bigcup_{g \in G} gH \end{align}</math></div>
+<ul>
+<li>The number of left cosets of <span class="math-inline"><math>H</math></span> is the index <span class="math-inline"><math>[G : H]</math></span> and the number of elements in <span class="math-inline"><math>gH</math></span> is equal to the number of elements in <span class="math-inline"><math>H</math></span>. So:</li>
+</ul>
+<div style="text-align: center;"><math>\begin{align} \quad \mid G \mid = [G : H] \mid H \mid \end{align}</math></div>
+<ul>
+<li>Therefore <span class="math-inline"><math>\mid H \mid</math></span> divides <span class="math-inline"><math>\mid G \mid</math></span>, i.e., the number of elements in any subgroup <span class="math-inline"><math>(H, \cdot)</math></span> of a finite group <span class="math-inline"><math>(G, \cdot)</math></span> must divide the number of elements in <span class="math-inline"><math>(G, \cdot)</math></span>. <span class="math-inline"><math>\blacksquare</math></span></li>
+</ul>
+===Corollaries to Lagrange's Theorem===
+<p>Recall that if <span class="math-inline"><math>(G, \cdot)</math></span> is a finite group and <span class="math-inline"><math>(H, \cdot)</math></span> is a subgroup then the number of elements in <span class="math-inline"><math>H</math></span> must divide the number of elements in <span class="math-inline"><math>G</math></span> and moreover:</p>
+<div style="text-align: center;"><math>\begin{align} \quad \mid G \mid = [G : H] \mid H \mid \end{align}</math></div>
+<p>We will now prove some amazing corollaries relating to Lagrange's theorem.</p>
+<blockquote style="background: white; border: 1px solid black; padding: 1em;">
+<td><strong>Corollary 1:</strong> Let <span class="math-inline"><math>(G, \cdot)</math></span> be a finite group and let <span class="math-inline"><math>(H, \cdot)</math></span> be a subgroup. Then the index of <span class="math-inline"><math>H</math></span> is <span class="math-inline"><math>G</math></span> is given by <span class="math-inline"><math>\displaystyle{[G : H] = \frac{\mid G \mid}{\mid H \mid}}</math></span>.</td>
+</blockquote>
+<ul>
+<li><strong>Proof:</strong> Rearrange the formula in the proof of Lagrange's theorem. <span class="math-inline"><math>\blacksquare</math></span></li>
+</ul>
+<blockquote style="background: white; border: 1px solid black; padding: 1em;">
+<td><strong>Corollary 2:</strong> Let <span class="math-inline"><math>(G, \cdot)</math></span> be a finite group and let <span class="math-inline"><math>g \in G</math></span>. Then the order of the element <span class="math-inline"><math>g</math></span> must divide <span class="math-inline"><math>\mid G \mid</math></span>.</td>
+</blockquote>
+<p><em>The order of an element <span class="math-inline"><math>g</math></span> is the smallest positive integer <span class="math-inline"><math>n</math></span> such that <span class="math-inline"><math>g^n = e</math></span> (where of course, <span class="math-inline"><math>e</math></span> is the identity element for the group.</em></p>
+<ul>
+<li><strong>Proof:</strong> The order of the element <span class="math-inline"><math>g</math></span> is the smallest positive integer <span class="math-inline"><math>n</math></span> such that <span class="math-inline"><math>g^n = e</math></span>. The generated subgroup <span class="math-inline"><math>(<g>, \cdot)</math></span> is such that <span class="math-inline"><math>\mid <g> \mid = n</math></span>. So by Lagrange's theorem <span class="math-inline"><math>n = \mid < g > \mid</math></span> must divide <span class="math-inline"><math>\mid G \mid</math></span>, i.e., the order of <span class="math-inline"><math>g</math></span> must divide <span class="math-inline"><math>\mid G \mid</math></span>.</li>
+</ul>
+<blockquote style="background: white; border: 1px solid black; padding: 1em;">
+<td><strong>Corollary 3:</strong> Let <span class="math-inline"><math>(G, \cdot)</math></span> be a finite group of order <span class="math-inline"><math>\mid G \mid = p</math></span> where <span class="math-inline"><math>p</math></span> is a prime number. Then <span class="math-inline"><math>G</math></span> is a cyclic group.</td>
+</blockquote>
+<ul>
+<li><strong>Proof:</strong> Let <span class="math-inline"><math>g \in G</math></span> be any non-identity element in <span class="math-inline"><math>G</math></span> (which exists since <span class="math-inline"><math>p \geq 2</math></span>). By Lagrange's Theorem, the subgroup <span class="math-inline"><math>(<g>, \cdot)</math></span> must be such that <span class="math-inline"><math>\mid <g> \mid</math></span> divides <span class="math-inline"><math>\mid G \mid = p</math></span>. But the only positive divisors of <span class="math-inline"><math>p</math></span> are <span class="math-inline"><math>1</math></span> and <span class="math-inline"><math>p</math></span>.</li>
+</ul>
+<ul>
+<li>So if <span class="math-inline"><math>\mid <g> \mid = 1</math></span> then <span class="math-inline"><math>g = e</math></span> which is a contradiction since we assumed that <span class="math-inline"><math>g</math></span> is not the identity for the group. So <span class="math-inline"><math>\mid <g> \mid = p</math></span>, i.e., <span class="math-inline"><math>G = <g></math></span>. So <span class="math-inline"><math>G</math></span> is a cyclic group.</li>
+</ul>
+<blockquote style="background: white; border: 1px solid black; padding: 1em;">
+<td><strong>Corollary 4:</strong> Let <span class="math-inline"><math>(G, \cdot)</math></span> be a finite group and let <span class="math-inline"><math>(H, \cdot)</math></span> and <span class="math-inline"><math>(I, \cdot)</math></span> be subgroups of <span class="math-inline"><math>G</math></span> such that <span class="math-inline"><math>I \subseteq H \subseteq G</math></span>. Then <span class="math-inline"><math>[G : I] = [G : H][H : I]</math></span>.</td>
+</blockquote>
+<ul>
+<li><strong>Proof:</strong> By Lagrange's theorem we have that:</li>
+</ul>
+<div style="text-align: center;"><math>\begin{align} \quad [G:I] & = \frac{\mid G \mid}{\mid I \mid} = \frac{\mid G \mid}{\mid H \mid} \cdot \frac{\mid H \mid}{\mid I \mid} = [G : H] [H : I] \quad \blacksquare \end{align}</math></div>
 == Licensing ==
 Content obtained and/or adapted from:
 * [http://mathonline.wikidot.com/left-and-right-cosets-of-subgroups Left and Right Cosets of Subgroups, mathonline.wikidot.com] under a CC BY-SA license
+* [http://mathonline.wikidot.com/the-set-of-left-right-cosets-of-a-subgroup-partitions-the-wh The Set of Left (Right) Cosets of a Subgroup Partitions the Whole Group, mathonline.wikidot.com] under a CC BY-SA license
+* [http://mathonline.wikidot.com/the-number-of-elements-in-a-left-right-coset The Number of Elements in a Left (Right) Coset, mathonline.wikidot.com] under a CC BY-SA license
+* [http://mathonline.wikidot.com/the-index-of-a-subgroup The Index of a Subgroup, mathonline.wikidot.com] under a CC BY-SA license
+* [http://mathonline.wikidot.com/lagrange-s-theorem Lagrange's Theorem, mathonline.wikidot.com] under a CC BY-SA license
+* [http://mathonline.wikidot.com/corollaries-to-lagrange-s-theorem Corollaries to Lagrange's Theorem, mathonline.wikidot.com] under a CC BY-SA license

Difference between revisions of "Cosets and Lagrange’s Theorem"

Latest revision as of 15:34, 17 November 2021

Contents

Left and Right Cosets of Subgroups

The Set of Left (Right) Cosets of a Subgroup Partitions the Whole Group

The Number of Elements in a Left (Right) Coset

The Index of a Subgroup

Lagrange's Theorem

Corollaries to Lagrange's Theorem

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