Problem:Prove or provide a counter example.

If H and K are subgroups of a group G, then is a subgroup of G.

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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?