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(4) Let (G, *) be a group and assume that z, a, b € G. Let also that c = x *a*r¹ and d = x * b*x-¹. Show that a * b = b* a if and only if cd=d*c.

(4) Let (G, *) be a group and assume that z, a, b € G. Let also that c = x *a*r¹ and d = x * b*x-¹. Show that a * b = b* a if and only if cd=d*c.

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter4: More On Groups
Section4.6: Quotient Groups
Problem 30E: Let G be a group with center Z(G)=C. Prove that if G/C is cyclic, then G is abelian.
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(4) Let (G, *) be a group and assume that z, a, b € G. Let also that c = x *a*x-¹ and d = x *b*x-¹. Show that a*b = b * a if and only if c*d=d* c.
Transcribed Image Text:(4) Let (G, *) be a group and assume that z, a, b € G. Let also that c = x *a*x-¹ and d = x *b*x-¹. Show that a*b = b * a if and only if c*d=d* c.
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