Exercise 3.2.12 (Bijection between the set of left cosets and the set of right cosets)

Question

Let $H \leq G$. Prove that the map $x \mapsto x^{-1}$ sends each left coset of $H$ in $G$ onto a right coset of $H$ and gives a bijection between the set of left cosets and the set of right cosets of $H$ in $G$ (hence the number of left cosets of $H$ in $G$ equals the number of right cosets).

Richard Ganaye · Accepted Answer

\begin{proof} 
Consider the map from the set of left cosets  relative to $H$ to the set of right cosets defined by
$$
\varphi \ 
\left\{
\begin{array}{ccl}
\{g H \mid g \in G\} & \mapsto & \{Hg \mid g \in G\}\
xH & \mapsto &Hx^{-1}
\end{array}
ight.
$$
Then
\begin{itemize}
\item $\varphi$ is well defined: If $x H = x' H$, where $x, x' \in G$, then $x^{-1} x' \in H$, thus $x^{-1} \in H x'^{-1}$, so $H x^{-1} = Hx'^{-1}$.

\item $\varphi$ is injective: If $\varphi(xH) = \varphi(yH)$, where $x, y \in G$, then $H x^{-1} = Hy^{-1}$, therefore $y^{-1} x \in H$, so $x \in y H$ and $xH = yH$.

\item $\varphi$ is surjective: If $Hy$ is any right coset, then $Hy = \varphi(y^{-1}H)$.
\end{itemize}
So $\varphi$ is bijective, hence the number of left cosets of $H$ in $G$ equals the number of right cosets.
\end{proof}

Exercise 3.2.12 (Bijection between the set of left cosets and the set of right cosets)

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Exercise 3.2.12 (Bijection between the set of left cosets and the set of right cosets)

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