Answered: Let Z denote the centre of a group G.…

Let Z denote the centre of a group G. Then show that z∈Z iff N(a)=G. Also show that if G is finite, then a∈Z iff o(N(a))=o(G).

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter3: Groups

Section3.2: Properties Of Group Elements

Problem 18E: Let a and b be elements of a group G. Prove that G is abelian if and only if (ab)2=a2b2.

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Question

Let Z denote the centre of a group G. Then show that z∈Z iff N(a)=G. Also show that if G is finite, then

a∈Z iff o(N(a))=o(G).

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Step 1: If z ∈ Z, then N(a) = G

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Step 2: If N(a) = G, then z ∈ Z

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Step 3: if a is in Z, then o(N(a)) = o(G).

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Step 4: if o(N(a)) = o(G), then a is in Z

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