Let a and b be elements of a group G. We define the conjugacy relation a ~ b by the following definition: a ~ b iff a = gbg-1 for some g e G. Prove that this conjugacy relation is an equivalence relation on G and conclude that conjugacy classes partition G.

Let a and b be elements of a group G. We define the conjugacy relation a ~ b by the following definition: a ~ b iff a = gbg-1 for some g e G. Prove that this conjugacy relation is an equivalence relation on G and conclude that conjugacy classes partition G.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter3: Groups

Section3.4: Cyclic Groups

Problem 30E

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Question

Let a and b be elements of a group G. We define the conjugacy relation a ~ b by the following definition: a ~ b iff

a = gbg^-1 for some g e G. Prove that this conjugacy relation is an equivalence relation on G and conclude that conjugacy classes partition G.

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