Let $S$ be a subset of a group $G$ that contains the identity element $1$ and such that the left cosets $aS$ with $a$ in $G$, partition $G$.Prove that $S$ a is a subgroup of $G$.
For $h$ in $S$, If I show that $hS=S$, then that would imply that $S$ is closed.
Now $hS$ is a partiton of $S$ and contains $h$ since $1$ is in $S$. Also $h$ is in $S$. Hence $h \in S\cap hS$. Moreover both of these are partitions and two partitions are either disjoint or equal. Hence $S=hS$ which says that $S$ is closed.
Does this seem alright??