Finite Group: Prime Order Property

Date: 02/11/2003 at 01:43:10
From: Nura
Subject: Group Theory

Suppose that G is a finite group with the property that every 
nonidentity element has prime order. If Z(G) is not trivial, prove 
that every nonidentity element of G has the same order.

Date: 02/11/2003 at 02:31:27
From: Doctor Jacques
Subject: Re: Group Theory

Hi Nura,

If Z(G) is not trivial, let us pick an element z <> e in Z(G). By 
hypothesis, z has prime order, say p.

We will show that every non-identity element has order p.

Let x be any element (<> e). By hypothesis, x has order q (a prime).

Consider the element y = xz.

As z is in the center, y^n = (xz)^n = x^n*z^n.

If q is not equal to p, can you prove that the order of y is pq ? 

Note that this will prove the theorem, since then the order of y 
would not be prime, contrary to the hypothesis.

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

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