Suppose that ¢ : G → G' is a group homomorphism and there is a group homomorphism (1) (a) v : G' → G such that ý o o = idG, where idg is the identity map on G. Show that o is injective. Show that the converse of part (a) is not true as follows: Consider the map ø : Z3 → S3 (b) given by $(0) = €, $(1) = (132), (2) = (123). %3D Show that o is injective but has no left-inverse, i.e there does not exist a homomorphism y : S3 → Z3 such that y o ¢ = idzą, where idz, is the identity map on Z3.
Suppose that ¢ : G → G' is a group homomorphism and there is a group homomorphism (1) (a) v : G' → G such that ý o o = idG, where idg is the identity map on G. Show that o is injective. Show that the converse of part (a) is not true as follows: Consider the map ø : Z3 → S3 (b) given by $(0) = €, $(1) = (132), (2) = (123). %3D Show that o is injective but has no left-inverse, i.e there does not exist a homomorphism y : S3 → Z3 such that y o ¢ = idzą, where idz, is the identity map on Z3.
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 33E
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