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 $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3 \mathbb{Z}}$ with representative $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2}$ 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 $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2 + 3\mathbb{Z} = 3\mathbb{Z} + 2}$ . But in general, is $gH=Hg$ 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 $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (G, \cdot)}$ 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 $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (S_3, \circ)}$ . Let $G=\{\epsilon ,(12)\}$ . Then $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (G, \circ)}$ 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 $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G \subset S_3}$ , $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G}$ is closed under $\circ$ , and $\epsilon ^{-1}=\epsilon$ , $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (12)^{-1} = (12)}$ (since $(12)$ 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 $Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G}$ 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 (13)G = \{ (13) \circ h : h \in G \} = \{ (13) \circ \epsilon, (13) \circ (12) \} = \{ (13), (123) \} \end{align}}

And the right 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:

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

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 $g\in G$ , $gH=Hg$ .

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