Proving H as a Subgroup of G: Using the Abelian Property

[iamalexalright]

Homework Statement

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 I am 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 I am 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!

 

[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.