33. Suppose that G is a group with the property that for every choiceof elements in G, axb = cxd implies ab = cd. Prove that G isAbelian. ("Middle cancellation" implies commutativity.)

Sol:- Let a, b be arbitrary elements of G. We want to show that ab = ba.

Answered: 33. Suppose that G is a group with the…

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 16E: Suppose ab=ca implies b=c for all elements a,b, and c in a group G. Prove that G is abelian.

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33. Suppose that G is a group with the property that for every choice of elements in G, axb = cxd implies ab = cd. Prove that G is Abelian. ("Middle cancellation" implies commutativity.)