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/
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