Exercise 3.2.7 ($a\sim b\iff b^{-1}a \in H$)

Question

Let $H \leq G$ and define the relation $\sim$ on $G$ by $a\sim b$ if and only if $b^{-1}a \in H$. prove that $\sim$ is an equivalence relation and describe the equivalence class of each $a \in G$. Use this to prove Proposition 4.

Richard Ganaye · Accepted Answer

\begin{proof} 
By definition, for all $a, b \in G$,
$$a \sim b \iff b^{-1}a \in H.$$
Then, for all $a,b,c \in G$,
\begin{enumerate}
\item[R.] $a^{-1}a = 1 \in H$, therefore $a \sim a$.
\item[S.] If $a \sim b$ then $b^{-1} a \in H$, therefore $a^{-1} b = (b^{-1}a)^{-1} \in H$, so $b \sim a$.
\item[T.] If $a \sim b$ and $b \sim c$, then $b^{-1}a \in H$ and $c^{-1} b \in H$, therefore  $ c^{-12} a = (c^{-1} b)(b^{-1} a) \in H$, so $a \sim c$.
\end{enumerate}
This shows that $\sim$ is an equivalence relation.

\bigskip
Let $a \in G$.
If $x \in \overline{a}$, then $x \sim a$, thus $a^{-1} x \in H$, so $x \in aH$.

Conversely, if $x \in aH$, then $x= a h$ for some $h \in H$, therefore $a^{-1}x = h \in H$, so $x \sim a$ and $x \in \overline{a}$. 

This shows that the class of $a \in G$ is $$\overline{a} = aH,$$ the left coset of representative $a$.

Since the set of classes is a partition of $G$, the set of left cosets is a partition of $G$. Furthermore
$$u H = vH \iff \overline{u} = \overline{v} \iff u \sim v \iff v^{-1} u \in N,$$
and in particular $uH = vH$ if and only $u \sim v$, i.e., if and only if $u$ and $v$ are representatives of the same coset. This proves Proposition 4 p. 80.
\end{proof}

Exercise 3.2.7 ($a\sim b\iff b^{-1}a \in H$)

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Exercise 3.2.7 ($a\sim b\iff b^{-1}a \in H$)

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