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1. Let G be a group, and let H and K be subgroups of G. Show that if KG, then HK/K ≈ H/(HK). This is known as the Second Isomorphism Theorem. (Hint: Let a: H → HK/K be the composition of the inclusion map H → HK and the coset map HK → HK/K. Show that a is a surjective homomorphism with kernel HK.)

1. Let G be a group, and let H and K be subgroups of G. Show that if KG, then HK/K ≈ H/(HK). This is known as the Second Isomorphism Theorem. (Hint: Let a: H → HK/K be the composition of the inclusion map H → HK and the coset map HK → HK/K. Show that a is a surjective homomorphism with kernel HK.)

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter3: Groups
Section3.5: Isomorphisms
Problem 5E
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1. Let G be a group, and let H and K be subgroups of G. Show that if K ◄ G, then HK/K ≈ H/(HK). This is known as the Second Isomorphism Theorem. (Hint: Let a: H → HK/K be the composition of the inclusion map H → HK and the coset map HK → HK/K. Show that a is a surjective homomorphism with kernel HnK.)
Transcribed Image Text:1. Let G be a group, and let H and K be subgroups of G. Show that if K ◄ G, then HK/K ≈ H/(HK). This is known as the Second Isomorphism Theorem. (Hint: Let a: H → HK/K be the composition of the inclusion map H → HK and the coset map HK → HK/K. Show that a is a surjective homomorphism with kernel HnK.)
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