Cosets and Lagrange’s Theorem

Left and Right Cosets of Subgroups

Definition: Let $(G,\cdot )$ be a group and let $(H,\cdot )$ be a subgroup. Let $g\in G$. Then the Left Coset of $H$ with Representative $g$ is the set $gH=\{gh:h\in H\}$. The Right Coset of $H$ with Representative $g$ is the set $Hg=\{hg:h\in H\}$.

When the operation symbol “$+$” is used instead of $\cdot$ we often denote the left and right cosets of $H$ with representation $g$ with the notation $g+H$ and $H+g$ respectively.

For example, consider the group $(\mathbb {Z} ,+)$ and the subgroup $(3\mathbb {Z} ,+)$. Consider the element $2\in \mathbb {Z}$. Then the left coset of $3\mathbb {Z}$ with representative $2$ is:

${\begin{aligned}\quad 2+3\mathbb {Z} =\{2+h:h\in 3\mathbb {Z} \}=\{...,-1,2,5,...\}\end{aligned}}$

And the right coset of $3\mathbb {Z}$ with representative $2$ is:

${\begin{aligned}\quad 3\mathbb {Z} +2=\{h+2:h\in 3\mathbb {Z} \}=\{...,-1,2,5,...\}\end{aligned}}$

In this particular example we see that $2+3\mathbb {Z} =3\mathbb {Z} +2$. But in general, is $gH=Hg$ for a given subgroup $(H,\cdot )$ of $(G,\cdot )$ and for $g\in G$? 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 $G=\{\epsilon ,(12)\}$. Then $(G,\circ )$ is a subgroup of $(S_{3},\circ )$ since $G\subset S_{3}$, $G$ is closed under $\circ$, and $\epsilon ^{-1}=\epsilon$, $(12)^{-1}=(12)$ (since $(12)$ is a transposition). Now consider the element $(13)\in S_{3}$. Then the left coset of $G$ with representative $(13)$ 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 $G$ with representative $(13)$ is:

${\begin{aligned}\quad G(13)=\{h\circ (13):h\in G\}=\{\epsilon \circ (13),(12)\circ (13)\}=\{(13),(132)\}\end{aligned}}$

We note that $(123)\neq (132)$ and so $(13)G\neq G(13)$!

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 $(G,\cdot )$ be a group and let $(H,\cdot )$ be a subgroup. If $(G,\cdot )$ is abelian then for all $g\in G$, $gH=Hg$.

• Proof: Let $g\in G$. If $G$ is abelian then for all $h\in G$ (and hence for all $h\in H$) we have that $g\cdot h=h\cdot g$. So:

${\begin{aligned}\quad gH=\{g\cdot h:h\in H\}=\{h\cdot g:h\in H\}=Hg\quad \blacksquare \end{aligned}}$

Proposition 2: Let $(G,\cdot )$ be a group, $(H,\cdot )$ a subgroup, and $g_{1},g_{2}\in G$. Then the following statements are equivalent:

a) $g_{1}H=g_{2}H$.
b) $Hg_{1}^{-1}=Hg_{2}^{-1}$.
c) $g_{1}H\subseteq g_{2}H$.
d) $g_{1}\in g_{2}H$.

e) $g_{1}^{-1}g_{2}\in H$.

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

Recall that if $(G,\cdot )$ is a group, $(H,\cdot )$ is a subgroup, and $g\in G$ then the left coset of $H$ with representative $g$ is the set:

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

The right coset of $H$ with representative $g$ is the set:

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

We will now look at a nice theorem which tells us that the set of all left cosets of a subgroup $(H,\cdot )$ actually partitions $(G,\cdot )$. The proof below can be mirrored to analogously show that the set of all right cosets of a subgroup $(H,\cdot )$ also partitions $(G,\cdot )$.

Theorem 1: Let $(G,\cdot )$ be a group and let $(H,\cdot )$ a subgroup. Then the set of all left cosets of $(H,\cdot )$ partitions $(G,\cdot )$.

Recall that a partition of a set $A$ is a collection of nonempty subsets of $A$ that are pairwise disjoint and whose union is all of $A$.

• Proof: We first show that any two distinct left cosets of $H$ are disjoint. Let $g_{1},g_{2}\in G$, $g_{1}\neq g_{2}$ and assume the left cosets $g_{1}H$ and $g_{2}H$ are distinct. Suppose that $g_{1}H\cap g_{2}H\neq \emptyset$. Then there exists an $x\in g_{1}H\cap g_{2}H$. So $x\in g_{1}H$ and $x\in g_{2}H$ and there exists $h_{1},h_{2}\in H$ such that:

${\begin{aligned}\quad x=g_{1}h_{1}\quad (*)\end{aligned}}$

${\begin{aligned}\quad x=g_{2}h_{2}\quad (**)\end{aligned}}$

• So using $(*)$ and $(**)$ we see that $g_{1}h_{1}=g_{2}h_{2}$. So $g_{1}=g_{2}h_{2}h_{1}^{-1}$. But $h_{2}h_{1}^{-1}\in H$ since $(H,\cdot )$ is a group and is hence closed under $\cdot$. So $g_{1}\in g_{2}H$. But this means that $g_{1}H=g_{2}H$, a contradiction since $g_{1}H$ and $g_{2}H$ distinct. So the assumption that $g_{1}H\cap g_{2}H\neq \emptyset$ was false. So:

${\begin{aligned}\quad g_{1}H\cap g_{2}H=\emptyset \end{aligned}}$

• Now if $i$ is the identity element for $(G,\cdot )$ then $i\in H$ since $(H,\cdot )$ is a subgroup and must contain the identity element. So $g=gi\in gH$ for all $g\in G$. So:

${\begin{aligned}\quad G=\bigcup _{g\in G}gH\end{aligned}}$

• Therefore the left cosets of $(H,\cdot )$ partition $(G,\cdot )$.

The Number of Elements in a Left (Right) Coset

Recall that if $(G,\cdot )$ is a group, $(H,\cdot )$ is a subgroup, and $g\in G$ then the left coset of $H$ with representative $g$ is defined as:

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

The right coset of $H$ with representative $g$ 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 $H$. This seems rather obvious since $gH$ contains elements of the form $gh$ where we range through all of $h$. So $gH$ has at most the same number of elements in $H$. Of course, $gH$ may have less elements if $gh_{1}=gh_{2}$ 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 $(G,\cdot )$ 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 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, $\mid Hg\mid =\mid H\mid$.

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 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 H} by:

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

• We will show that $f$ is bijective. First we show that $f$ is injective. Let $h_{1},h_{2}\in H$ and assume that $f(h_{1})=f(h_{2})$. Then we have that:

${\begin{aligned}\quad gh_{1}=gh_{2}\end{aligned}}$

• By left cancellation this implies that $h_{1}=h_{2}$ and so $f$ 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 gh \in gH} . 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 f(h) = gh} (somewhat trivially) so $f$ 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 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 = \mid gH \mid} so the number of elements in the left coset $gH$ 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 $(G,\cdot )$ is a group 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, \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 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} .

Theorem 1: 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. Then 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} .

• Proof: Let $L_{H}$ denote 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} 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 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 : L_H \to R_H} for all $gH\in L_{H}$ 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 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 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 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 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 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 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 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 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 $H$ 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} . $\blacksquare$

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 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 : H]} is defined as the number of left (or 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} .

For example, consider the symmetric 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 (S_3, \circ)} 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 = \{ \epsilon, (12) \}} . Let's find the index 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 ]} . Note that $S_{3}=\{\epsilon ,(12),(13),(23),(123),(132)\}$. So 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 H} 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 S_3} 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 \epsilon H = \{ \epsilon \circ \epsilon, \epsilon \circ (12) \} = \{ \epsilon, (12) \} \\ \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 (12)H= \{ (12) \circ \epsilon, (12) \circ (12) \} = \{ (12), \epsilon \} \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 (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}}

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 (123)H = \{ (123) \circ \epsilon, (123) \circ (12) \} = \{ (123), (13) \} \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 (132)H = \{ (132) \circ \epsilon, (132) \circ (12) \} = \{ (132), (23) \} \end{align}}

So the set of left cosets of $H$ 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 \begin{align} \quad \{\{ \epsilon, (12) \}, \{ (13), (123) \}, \{ (23), (132) \} \} \end{align}}

There are three such cosets so $[S_{3}:H]=3$.

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 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}, +)} . 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}}

${\begin{aligned}\quad 1+3\mathbb {Z} =\{...,-2,1,4,...\}\end{aligned}}$

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 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 $(G,\cdot )$ 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 $g_{1},g_{2}\in G$ 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, 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 = \emptyset} and:

${\begin{aligned}\quad G=\bigcup _{g\in G}gH\end{aligned}}$

• 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 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:

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

• 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 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 number of elements in any 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 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 $(H,\cdot )$ 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 $\displaystyle {[G:H]={\frac {\mid G\mid }{\mid H\mid }}}$.

• Proof: Rearrange the formula in the proof of 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 \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 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} such that $g^{n}=e$ (where of course, 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 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 $\mid G\mid$, 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 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} .

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 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} 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 $G$ 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 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} (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 $\mid <g>\mid$ 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 $1$ 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 $\mid <g>\mid =1$ 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 = 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 $G$ 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