Let G be a group and H be a non-empty subset of G. We define H^2 as the set of elements of G which can be written in the form h_1 h_2, with h_1 ,h_2∈H.
Let H be finite. Prove that H is a subgroup if and only if H^2=H.
Proof. Assume that H is a subgroup. Then it is closed under multiplication, and hence H^2 is a subset of H. For the converse, let us assume that h∈H. Since h=he, and e∈H, we conclude that H is a subset of H^2, and hence ...
The solution provides the criterion for a finite non-empty subset of a group to be a subgroup.