Let G be a group. The centre of G is defined as the set Z(G) = {g ∈ G : gx = xg for all x ∈ G}. Show that Z(G) is a normal subgroup of G. Further, show that if the quotient group G/Z(G) is cyclic, then the group G is Abelian.

Let G be a group. The centre of G is defined as the set Z(G) = {g ∈ G : gx = xg for all x ∈ G}. Show that Z(G) is a normal subgroup of G. Further, show that if the quotient group G/Z(G) is cyclic, then the group G is Abelian.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter4: More On Groups

Section4.4: Cosets Of A Subgroup

Problem 29E: Let be a group of order , where and are distinct prime integers. If has only one subgroup of...

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Let G be a group. The centre of G is defined as the set

Z(G) = {g ∈ G : gx = xg for all x ∈ G}.

Show that Z(G) is a normal subgroup of G. Further, show that if the quotient group G/Z(G) is cyclic, then the group G is Abelian.

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