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G is an abelian group

Let [itex]H = {x \in G : x = x^{-1}[/itex]

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!

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# Homework Help: Proving subgroup of group

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