Suppose that G is a finite group with the property that every nonidentityelement has prime order (for example, D3 and D5). If Z(G)is not trivial, prove that every nonidentity element of G has thesame order.

Suppose that G is a finite group with the property that every nonidentityelement has prime order (for example, D3 and D5). If Z(G)is not trivial, prove that every nonidentity element of G has thesame order.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter3: Groups

Section3.4: Cyclic Groups

Problem 39E

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Question

Suppose that G is a finite group with the property that every nonidentity
element has prime order (for example, D₃ and D₅). If Z(G)
is not trivial, prove that every nonidentity element of G has the
same order.

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