question 3 (1)Let G be a group. Show that the map f: G→ G given by f(x) = x−¹ isan automorphism of G if and only if G is abelian.(2)Let N be a normal subgroup of G such that |G/N is finite. Suppose g = Gand the order of g is coprime to |G/N. Show that g € N.(3)Suppose N is a normal subgroup of G such that N [G, G] is the trivialsubgroup. Show that N is contained in the center Z of G.

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Answered: (3) Suppose N is a normal subgroup of G…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter4: More On Groups

Section4.5: Normal Subgroups

Problem 28E: 28. For an arbitrary subgroup of the group , the normalizer of in is the set . a. Prove...

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(3) Suppose N is a normal subgroup of G such that N [G, G] is the trivial subgroup. Show that N is contained in the center Z of G.