Problem: Prove or provide a counter example.
If H and K are subgroups of a group G, then is a subgroup of G.
I had the set of generators 0 through 17 for sitting in front of me from a previous problem.
I saw that
So I said and .
Therefore, would be
But this isn't closed under the group operation. and
So am I correct in saying that will not always be a subgroup of G?
Even if I'm correct, I'm a little concerned. If I had not had the generator list in front of me, I wouldn't have immediately spotted the counterexample. Is there any mathematically concrete way to go about this?