Exercise 7.2. Let H be a normal subgroup of a group G with m := [G: H] < ∞.(1) Show that for a Є H for every a Є G.(2) Suppose H and the factor group G/H are both cyclic and gcd(|H|, |G/H|) = 1. Prove ordisprove that G is cyclic.

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Answered: Exercise 7.2. Let H be a normal…

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 21E

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Exercise 7.2. Let H be a normal subgroup of a group G with m := [G: H] < ∞. (1) Show that for a Є H for every a Є G. (2) Suppose H and the factor group G/H are both cyclic and gcd(|H|, |G/H|) = 1. Prove or disprove that G is cyclic.