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Let G be an abelian group, Ha subgroup of G, and n a fixed positive integer. (a) Let Hn = (aEGla" EH}. Prove that H, is a subgroup of G. (b) Show part (a) is false if G is not abelian as follows: Take G (S3, o), H = ((1)} (the identity subgroup), and n = 2 %3D Show H2 = {y E S3Iy2 (1)} is not a subgroup. %3D

Let G be an abelian group, Ha subgroup of G, and n a fixed positive integer. (a) Let Hn = (aEGla" EH}. Prove that H, is a subgroup of G. (b) Show part (a) is false if G is not abelian as follows: Take G (S3, o), H = ((1)} (the identity subgroup), and n = 2 %3D Show H2 = {y E S3Iy2 (1)} is not a subgroup. %3D

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter4: More On Groups
Section4.4: Cosets Of A Subgroup
Problem 30E: Let G be an abelian group of order 2n, where n is odd. Use Lagranges Theorem to prove that G...
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Let G be an abelian group, Ha subgroup of G, and n a fixed positive integer. (a) Let Hn = (a EGla" EH}. Prove that Hn is a subgroup of G. %3D (b) Show part (a) is false if G is not abelian as follows: Take G = (S3, o), H = {(1)} (the identity subgroup), and n = 2 %3D Show H2 {y E S3 Iy? = (1)} is not a subgroup. %3D
Transcribed Image Text:Let G be an abelian group, Ha subgroup of G, and n a fixed positive integer. (a) Let Hn = (a EGla" EH}. Prove that Hn is a subgroup of G. %3D (b) Show part (a) is false if G is not abelian as follows: Take G = (S3, o), H = {(1)} (the identity subgroup), and n = 2 %3D Show H2 {y E S3 Iy? = (1)} is not a subgroup. %3D
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