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(5) Show that in a group G of odd order, the equation x² = a has a unique solution for all a e G.

(5) Show that in a group G of odd order, the equation x² = a has a unique solution for all a e 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 Q5, thanks

Suppose that ¢ : G → G' is a group homomorphism and there is a group homomorphism (1) (a) V : G' → G such that ý o ø = 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 o : Z3 → S3 $(0) = e, $(1) = (132), $(2) = (123). %3D Show that o is injective but has no left-inverse, i.e there does not exist a homomorphism b : S3 → Z3 such that y o ø = idz3, where idz, is the identity map on Z3. (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) (1 5 8 7)(1 3 4 ) E Sg. Is a an even or odd permutation? Justify your answer. Find a-1. Find |a|. Find a51. Suppose G is a group with order pq, where p and q are distinct prime numbers. If G has only one (4) subgroup of order p and only one subgroup of order q, prove that G is cyclic. (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 ý o ø = 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 o : Z3 → S3 $(0) = e, $(1) = (132), $(2) = (123). %3D Show that o is injective but has no left-inverse, i.e there does not exist a homomorphism b : S3 → Z3 such that y o ø = idz3, where idz, is the identity map on Z3. (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) (1 5 8 7)(1 3 4 ) E Sg. Is a an even or odd permutation? Justify your answer. Find a-1. Find |a|. Find a51. Suppose G is a group with order pq, where p and q are distinct prime numbers. If G has only one (4) subgroup of order p and only one subgroup of order q, prove that G is cyclic. (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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