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Date: 02/09/2003 at 23:49:51
From: Christina
Subject: Abstract Algebra

If the number of elements in a finite group G with identity e is even, 
show that there is at least one element g in G such that g <> e but 
g * g = e.

I was looking through my textbook and I came across this exercise and 
I was confused about how to prove it.

Date: 02/10/2003 at 03:25:41
From: Doctor Jacques
Subject: Re: Abstract Algebra

Hi Christina,

First, note that g*g = e is the same as g = g^(-1).

Let us pair the elements of G with their inverses. This will give 
three types of subsets.

(a) the subset {e}
(b) subsets of type {x, x^(-1)} for all x such that x and x^(-1) are 
    distinct.
(c) subsets of type {x} whenever x = x^(-1)

We must show that there is at least one subset of type (c).

Notice that these subsets constitute a partition of G (they are 
pairwise disjoint, and their union is G).

Notice also that the subset {e} will contribute one element, and each 
subset of type (b) will contribute two elements, i.e. the union of 
the subsets of type (b) will contribute an even number of elements.

You should be able to conclude the proof from here.

Please feel free to write back if you are still stuck.

- Doctor Jacques, The Math Forum
  http://mathforum.org/dr.math/