Show G is Abelian: Let G be a Finite Group w/ I

[playa007]

Homework Statement

Let G be a finite group and let I ={g in G: g^2 = e} \ {e} be its subset of involutions. Show that G is abelian if card(I) => (3/4)card(G).

Homework Equations

The Attempt at a Solution

I don't really know how to proceed with this problem and to make use of 3/4. I know that the set I is precisely the elements of order 2 in the group G and all elements within I U {e} commute; but I just don't know how to show the whole group commutes. Any help on how to proceed with the problem is highly appreciated. Thanks.

 

[Dick]

I'll give you a hint while I'm trying to figure this one out myself. You don't know that I commutes, do you? It's perfectly possible that a^2=e and b^2=e and ab is not equal to ba, isn't it? If you HAVE figured out how to prove {e} U I is commutative, then you would be done. That makes it a subgroup. Use Lagrange's theorem.

 

[Dick]

For future reference, I think I finally got this one. Consider an element a in I. Now look at the |G| equations aG=G. Show if ab=c and b and c are also in I, then b commutes with a. Count how many equations in aG=G are of that form and draw a conclusion about the size of the centralizer subgroup of a. The rest is an easy exercise. That was bothering me.