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(a) Let eg be the identity element of G, let e be the identity element of H, and let g E G. Prove that o(eG) = eH and þ(g¯¹) = þ(g)¯¹. (b) Let G be a commutative group. Prove that the map : G → G defined by (g) = g² is a homomorphism. Give an example of a noncommutative group for which this map is not a homomorphism. (c) Same question as (b) for the map o(g) = g¯¹.

(a) Let eg be the identity element of G, let e be the identity element of H, and let g E G. Prove that o(eG) = eH and þ(g¯¹) = þ(g)¯¹. (b) Let G be a commutative group. Prove that the map : G → G defined by (g) = g² is a homomorphism. Give an example of a noncommutative group for which this map is not a homomorphism. (c) Same question as (b) for the map o(g) = g¯¹.

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter4: More On Groups
Section4.2: Cayley’s Theorem
Problem 12E: Find the right regular representation of G as defined Exercise 11 for each of the following groups....
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2.13. Let G and H be groups. A function : G→ H is called a (group) homomor- phism if it satisfies (91 * 92) = (91) * (92) for all 91,92 € G. (Note that the product g₁ ★ 92 uses the group law in the group G, while the prod- uct (91) * (92) uses the group law in the group H.) 108 Exercises (a) Let eg be the identity element of G, let e be the identity element of H, and let g E G. Prove that o(еG) = еH and o(g¯¹) = þ(g)−¹. (b) Let G be a commutative group. Prove that the map : G → G defined by (g) = g² is a homomorphism. Give an example of a noncommutative group for which this map is not a homomorphism. (c) Same question as (b) for the map ø(g) = g¯¹.
Transcribed Image Text:2.13. Let G and H be groups. A function : G→ H is called a (group) homomor- phism if it satisfies (91 * 92) = (91) * (92) for all 91,92 € G. (Note that the product g₁ ★ 92 uses the group law in the group G, while the prod- uct (91) * (92) uses the group law in the group H.) 108 Exercises (a) Let eg be the identity element of G, let e be the identity element of H, and let g E G. Prove that o(еG) = еH and o(g¯¹) = þ(g)−¹. (b) Let G be a commutative group. Prove that the map : G → G defined by (g) = g² is a homomorphism. Give an example of a noncommutative group for which this map is not a homomorphism. (c) Same question as (b) for the map ø(g) = g¯¹.
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