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.
Best Answer
A coset is a set while a group is a set together with a binary operation that satisfies some axioms. So, a coset is not a group since the binary operation is missing. Any question asking whether a given set is a group is a wrong question. If you meant to ask if a coset is a subgroup (of the obvious ambient group), then that can be answered negatively by noticing that the identity element, which must be an element of any subgroup, is not necessarily an element in a coset.