# Proving subgroup of group

1. Sep 20, 2011

### iamalexalright

1. The problem statement, all variables and given/known data
G is an abelian group
Let $H = {x \in G : x = x^{-1}$

Prove H is a subgroup of G.

I have two methods in my arsenal to do this (and Im writing them out additively just for ease):
1. Let a,b be in H. If a + b is in H AND -a is in H then H<G.
or
2.Let a,b be in H. if a-b is in H then H<G.

Solution:
If I use method one the 2nd part is given practically (if a is in H then a^-1 = a is certainly in H).

Then I need to show ab is in H. this is what Im struggling with... I feel (since it is given) I should use the fact that G is abelian but not sure where/how to do that!

2. Sep 20, 2011

### micromass

So you need to show that

$$ab=(ab)^{-1}$$

First, write out what $(ab)^{-1}$ is. Then us that a and b are in H.