Hi. I'm trying to prove the following:

Let be a finite group, and let . Show that the set has an even number of elements.

So, Here's my thoughts.

Let the number of all elements such that be so that there are elements in not equal to their own inverses. But, is a group and so there exists for every element a such that the identity element of . Thus, there are also inverses corresponding to the elements belonging to , but these elements also belong to since , Therefore, there are elements in . Therefore, there are an even number of elements in

Is this Right. The thing that concerns me is the fact that from this argument . Can someone help me phrase this better? I'm new to this subject and I've only recently became mature enough to construct proofs like this.