No chatgpt pls will upvote (b) (i) Let G be the set of all permutations of the positive integers. Let H be the subset of elementsof G that can be expressed as a product of a finite number of cycles. Prove that H is asubgroup of G.(ii) Let a and ẞ belongs to Sr. Prove that Baẞ-¹ and a are both even or both odd.3+2

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Answered: (b) (i) Let G be the set of all…

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 30E: 30. Prove statement of Theorem : for all integers .

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(b) (i) Let G be the set of all permutations of the positive integers. Let H be the subset of elements of G that can be expressed as a product of a finite number of cycles. Prove that H is a subgroup of G. (ii) Let a and ẞ belongs to Sr. Prove that Baẞ-¹ and a are both even or both odd. 3+2