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(d) Find o51 Suppose G is a group with order pq, where p and q are distinct prime numbers. If G has only one subgroup of order p and only one subgroup of order q, prove that G is cyclic. Show that in a group G of odd order, the equation x2 = a has a unique solution for all a € G.

(d) Find o51 Suppose G is a group with order pq, where p and q are distinct prime numbers. If G has only one subgroup of order p and only one subgroup of order q, prove that G is cyclic. Show that in a group G of odd order, the equation x2 = a has a unique solution for all a € G.

Advanced Engineering Mathematics
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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[Groups and Symmetries] How do you solve Q4, thanks

Suppose that ¢ : G → G' is a group homomorphism and there is a group homomorphism (1) (a) V : G' → G such that oo = idG', where idG' is the identity map on G'. Show that o is injective. (b) given by Show that the converse of part (a) is not true as follows: Consider the map ø : Z3 → S3 Ф(0) %— с, Ф(1) 3 (132), Ф(2) — (123). %3D Show that o is injective but has no left-inverse, i.e there does not exist a homomorphism ý : S3 → Z3 such that o o = idz3, where idz, is the identity map on S3. (2) of order 5 is in H. Let G be a group of order 100 that has a subgroup H of order 25. Prove that every element of G (3) Let α- (a) (b) (c) (d) (1 5 8 7)(1 3 4 6) e S9. Is a an even or odd permutation? Justify your answer. Find a-1. Find |a|. Find a51. (4) subgroup of order p and only one subgroup of order q, prove that G is cyclic. Suppose G is a group with order pq, where p and q are distinct prime numbers. If G has only one (5) Show that in a group G of odd order, the equation x2 = a has a unique solution for all a e G.
Transcribed Image Text:Suppose that ¢ : G → G' is a group homomorphism and there is a group homomorphism (1) (a) V : G' → G such that oo = idG', where idG' is the identity map on G'. Show that o is injective. (b) given by Show that the converse of part (a) is not true as follows: Consider the map ø : Z3 → S3 Ф(0) %— с, Ф(1) 3 (132), Ф(2) — (123). %3D Show that o is injective but has no left-inverse, i.e there does not exist a homomorphism ý : S3 → Z3 such that o o = idz3, where idz, is the identity map on S3. (2) of order 5 is in H. Let G be a group of order 100 that has a subgroup H of order 25. Prove that every element of G (3) Let α- (a) (b) (c) (d) (1 5 8 7)(1 3 4 6) e S9. Is a an even or odd permutation? Justify your answer. Find a-1. Find |a|. Find a51. (4) subgroup of order p and only one subgroup of order q, prove that G is cyclic. Suppose G is a group with order pq, where p and q are distinct prime numbers. If G has only one (5) Show that in a group G of odd order, the equation x2 = a has a unique solution for all a e G.
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