Here's the question:(adsbygoogle = window.adsbygoogle || []).push({});

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.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Subgroups-Stuck on a question

**Physics Forums | Science Articles, Homework Help, Discussion**