September 28th, 2007, 09:22 AM  #1 
Newbie Joined: Sep 2007 Posts: 9 Thanks: 0  finite group
Let G be a finite group of even order. Prove that G contains at least one element of order 2 ( that is, some c different from e with c^2=e).

September 28th, 2007, 12:08 PM  #2 
Global Moderator Joined: Dec 2006 Posts: 19,711 Thanks: 1805 
If p is any element other than e that's its own inverse, p = p^1, so pÂ² = e, so p is of order two. There's an even number of elements which are not their own inverse, since they can be counted in pairs, each one paired with its inverse. There's therefore (since the group is of even order) an even number of elements which are their own inverse. The identity, e, is one, so there are an odd number of others, each being of order two. 

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