For any subgroup \( H \) of \( G \), we can form an equivalence relation:
\[ x \sim_{R} y \iff xy^{-1} \in H \]
The partition created by this equivalence relation is the set of right cosets.
Or a (possibly different) equivalence relation:
\[ x \sim_{L} y \iff x^{-1}y \in H \]
The partition created by this equivalence relation is the set of left cosets.
Property
The equivalence relation:
\[ x \sim_{R} y \iff xy^{-1} \in H \]
Has the following important property:
\[
\begin{equation}
\forall g \in G, \quad x \sim_{R} y \implies xg \sim yg \tag{1}
\end{equation}
\]
A similar argument shows that the equivalence relation:
\[ x \sim_{L} y \iff x^{-1}y \in H \]
Has the property:
\[
\begin{equation}
\forall g \in G, \quad x \sim_{L} y \implies gx \sim gy \tag{2}
\end{equation}
\]
How are these properties related to quotient groups?
Connection to quotient groups
An equivalence relation that satisfies (1) and (2) forms a quotient that
is a group, whose operation is inherited from the original group.
Proof for right cosets
\( xy^{-1} \in H \) implies that \( x \) is in the coset \( Hy \) and
that \( x \) can be expressed as \( h_1 y \) for some \( h_1 \in H \).
Then, we see that for \( x \) we have \( xg = h_1 y g \) and
for \( y \) we have \( yg = e y g \). Thus, both \( xg \) and \( yg \)
are in the same right coset of \( H \), which is the requirement for equivalence.
Proof for right cosets (v2)
Proof.
\[
\begin{align*}
xy^{-1} \in H &\implies x(gg^{-1})y^{-1} \in H \\
&\implies (xg)(g^{-1}y^{-1}) \in H \\
&\implies (xg)(yg)^{-1} \in H \\
\end{align*}
\]
