# Changes

## Left cosets partition a group

, 00:04, 4 July 2011
Proof in form (2)
'''To prove''': For any $a \in G$, $a \sim a$.
'''Proof''': Clearly $e \in H$ (since $H$ is a subgroup). Hence, for any $a \in G$, $a = ae$, so $\! a \sim a$: $a$ is in its own left coset.
===Symmetry===
'''To prove''': For any $a,b \in G$ such that $\! a \sim b$, we have $\! b \sim a$.
'''Proof''': If $a = bh$, for some $h \in H$, then $b = ah^{-1}$. Since $h \in H$ and $H$ is a subgroup, $h^{-1} \in H$. Thus, if $a$ is in the left coset of $b$, then $b$ is in the left coset of $a$. In symbols, $a \sim b \implies b \sim a$.