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ab Define * on Q* by a * b =: Show that (Q*,*) is a group. 1.1 1.2 Let a, b be elements of a group G. Assume that a has order 5 and a'b = ba³. Prove that ab = ba. Let G be a group and Z(G) = {a e G: ax = xa for all x E G}. Show that Z(G)is 1.3 a normal subgroup of G. Let G be a group and let p:G → G be the map p(x) = x-1. (a) Prove that p is bijective. (b) Prove that p is an automorphism if G is abelian. Let a, ßE S, (Symmetric group), where a = (1,2)(4,5) and ß = (1,6,5,3,2). Verify that (aß)1 = B-1a-1 . 1.4 1.5

ab Define * on Q* by a * b =: Show that (Q*,*) is a group. 1.1 1.2 Let a, b be elements of a group G. Assume that a has order 5 and a'b = ba³. Prove that ab = ba. Let G be a group and Z(G) = {a e G: ax = xa for all x E G}. Show that Z(G)is 1.3 a normal subgroup of G. Let G be a group and let p:G → G be the map p(x) = x-1. (a) Prove that p is bijective. (b) Prove that p is an automorphism if G is abelian. Let a, ßE S, (Symmetric group), where a = (1,2)(4,5) and ß = (1,6,5,3,2). Verify that (aß)1 = B-1a-1 . 1.4 1.5

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter4: More On Groups
Section4.5: Normal Subgroups
Problem 13E: Exercise 8 states that every subgroup of an abelian group is normal. Give an example of a nonabelian...
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Define * on Q* by a * b = - ab Show that (Q*,*) is a group. 1.1 1.2 Let a, b be elements of a group G. Assume that a has order 5 and a³b = ba³. Prove that ab = ba. 1.3 Let G be a group and Z(G) = {a € G:ax = xa for all x e G}. Show that Z(G)is a normal subgroup of G. Let G be a group and let p: G → G be the map p(x) = x-1. (a) Prove that o is bijective. (b) Prove that p is an automorphism if G is abelian. Let a, ßE S, (Symmetric group), where a = (1,2)(4,5) and ß = (1,6,5,3,2). Verify that (aß)-1 = B-'a-1 . 1.4 1.5
Transcribed Image Text:Define * on Q* by a * b = - ab Show that (Q*,*) is a group. 1.1 1.2 Let a, b be elements of a group G. Assume that a has order 5 and a³b = ba³. Prove that ab = ba. 1.3 Let G be a group and Z(G) = {a € G:ax = xa for all x e G}. Show that Z(G)is a normal subgroup of G. Let G be a group and let p: G → G be the map p(x) = x-1. (a) Prove that o is bijective. (b) Prove that p is an automorphism if G is abelian. Let a, ßE S, (Symmetric group), where a = (1,2)(4,5) and ß = (1,6,5,3,2). Verify that (aß)-1 = B-'a-1 . 1.4 1.5
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