I ran into this question when reading Artin's Algebra book and tried to google the answer but seems it's too easy that I didn't get any.

My answer is: Not necessarily.

Proof:

From the definition of a left coset: $aH=\{ah\ |\ h\in H\}$, where $H$ is a subgroup of $G$, and $a\in G$. 

If we want $aH$ to be a group, it has to be a closure: $ah\_{1}ah\_{2}\in aH$, thus $h\_{1}ah\_{2}\in H$. And since $a\in G$ according to the assumption, $h\_{1}ah\_{2}\notin H$. So cosets are not necessarily groups.

Am I correct? Thanks.