September 28th, 2007, 10:22 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, 01:08 PM  #2 
Global Moderator Joined: Dec 2006 Posts: 20,099 Thanks: 1905 
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. 

Tags 
finite, group 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
example of a finite group....  rayman  Abstract Algebra  7  March 4th, 2012 06:41 AM 
Prove that this finite set is a group  nata  Abstract Algebra  2  October 30th, 2011 06:29 PM 
Let G be a finite group and h:G–>G an isomorphism, such that  johnmath  Abstract Algebra  2  November 27th, 2010 11:52 PM 
when is this factor/quotient group finite?  aptx4869  Abstract Algebra  0  September 26th, 2007 02:27 AM 
T'group finite (helpme please thanks)  sastra81  Abstract Algebra  1  January 9th, 2007 08:05 AM 