Q3) Prove or disprove1. If (G, *) is an abelian group, then the map : G→ G defined by y(x) = x-1is a homomorphism.2. If (G,*) is an abelian group, then Z(G) = G.3. (Z8, +) (D4, 0).=4. Any non-trivial group has at least 2 normal subgroups.

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Answered: Q3) Prove or disprove 1. If (G, *) is…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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46. Determine whether (Z, - {0}, 6 ) is it a group or not? Explain your answer?
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3.11 Let (G,) be a group such that x²-e for all x E G. Show that (G,.) is abelian. (Here x2 means xx.)
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. Prove that the set S0(2) = { : z,y } forms an abelian group with respect to multiplication.
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2. (a) Let be a subgroup of the center of G. Show that if G/N is a cyclic group, then G must be abelian. (b) Show that G/Z(G) cannot be a nontrivial cyclic group.
Let 0:250 Z15 be a group homomorphism with 0(x)=4x. Then, Ker(0)= * O (0, 10, 20, 30, 40} None of the choices (0, 5, 15, 25, 35, 45) O (0,15,30,45} acer
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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Q3) Prove or disprove 1. If (G, *) is an abelian group, then the map : G→ G defined by p(x): is a homomorphism. =x-1 2. If (G, *) is an abelian group, then Z(G) = G. 3. (Z8, +) (D4, 0). 4. Any non-trivial group has at least 2 normal subgroups.

Q3) Prove or disprove 1. If (G, ) is an abelian group, then the map : G→ G defined by p(x): is a homomorphism. =x-1 2. If (G, ) is an abelian group, then Z(G) = G. 3. (Z8, +) (D4, 0). 4. Any non-trivial group has at least 2 normal subgroups.