Cosets and Lagrange’s Theorem

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 $g\in G$. 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:

Failed to parse (MathML with SVG or PNG fallback (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)G = \{ (13) \circ h : h \in G \} = \{ (13) \circ \epsilon, (13) \circ (12) \} = \{ (13), (123) \} \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 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:

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

So, when exactly are the left and right cosets of a subgroup 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} 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 $g\in G$, $gH=Hg$.

• Proof: Let $g\in G$. 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} 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 Failed to parse (MathML with SVG or PNG fallback (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} . Then the following statements are equivalent:

a) $g_{1}H=g_{2}H$.
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) Failed to parse (MathML with SVG or PNG fallback (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 \subseteq g_2H} .
d) $g_{1}\in g_{2}H$.

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} .

Licensing

Content obtained and/or adapted from:

• Left and Right Cosets of Subgroups, mathonline.wikidot.com under a CC BY-SA license