- #1

hgj

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Prove that a nonempty subset H of a group G is a subgroup if for all x and y in H, the element xy^(-1) is also in H.

We have a theorem that says if G is a group and H is a nonempty subset of G, then H is a subgroup of G iff:

(1) H is closed

(2) if h is in H, then the inverse of h in G lies in H.

I know I need to use this theorem, and I have two ideas about how to go about it:

First, I think if I can prove y^(-1) is in H, then the two parts of the theorem follow from that. Unfortunately, I can't figure out how to do this.

My second thought is to prove the two parts separately. If I do this, my proof for closure is:

Assume xy is not in H. Then x is not in H or y is not in H. But this is a contradiction, so xy must be in H. Then closure is satisfied.

After this, though, I get stuck again when I try to prove part (2) about the inverse.