## Alternate characterization of cosets as equivalence classes

Let $G$ be a group and $H \leq G$. Define a 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 for each $a \in G$ the equivalence class $[a]$. Use this to prove the following proposition.

Proposition: Let $G$ be a group and $H \leq G$. Then (i) the set of left cosets of $H$ is a partition of $G$ and (ii) for all $u,v \in G$, $uH = vH$ if and only if $v^{-1} u \in H$.

1. $\sim$ is an equivalence:
1. Reflexive: $x^{-1}x = 1 \in H$, so that $x \sim x$ for all $x \in G$.
2. Symmetric: Suppose $x \sim y$. Then $y^{-1} x \in H$. Since $H$ is a subgroup, $(y^{-1}x)^{-1} = x^{-1} y \in H$, and we have $y \sim x$.
3. Transitive: Suppose $x \sim y$ and $y \sim z$. Then $y^{-1}x \in H$ and $z^{-1}y \in H$. Thus we have $y^{-1}x = h_1$ and $z^{-1}y = h_2$ for some $h_1,h_2 \in H$. Now $y = xh_1^{-1}$, and substituting yields $z^{-1}xh_1^{-1} = h_2$, hence $z^{-1}x = h_2h_1$. Thus $x \sim z$.

So $\sim$ is an equivalence.

2. Let $x \in G$ and suppose $y \sim x$. Then $x^{-1}y = h$ for some $h \in H$, hence $y = xh$. Thus $y \in xH$; so $[x] \subseteq xH$. Now suppose $y \in xH$. Then $y = xh$ for some $h \in H$, hence $x^{-1}y \in H$ and we have $y \sim x$, so that $xH \subseteq [x]$. Thus the $\sim$-equivalence classes of $G$ are precisely the left cosets of $H$.
3. Proof of Proposition 4: The left cosets of $H$ form the equivalence classes of a relation $\sim$ on $G$ defined by $x \sim y$ if and only if $y^{-1}x \in H$; the second conclusion of Proposition 4 follows trivially.