Let G be a finite group of order 56. a) Determine the possible number of Sylow 7-subgroups in G. Provide a detailed justification using Sylow's theorems. b) Assume that G has exactly one Sylow 7-subgroup. Prove that this Sylow 7-subgroup is normal in G, and subsequently show that G contains a normal subgroup of order 8. c) Using the results from parts (a) and (b), classify all possible group structures for G. Discuss whether G is necessarily abelian or non-abelian based on your classification. Let G be a finite non-abelian simple group of order 60. a) Determine all possible isomorphism types of G. Justify your conclusion using the classification of simple groups of small order. b) Compute the automorphism group Aut(G) of G. Provide a detailed analysis of its structure, including any inner and outer automorphisms. c) Prove that every automorphism of G is inner or describe the existence of outer automorphisms. Use this to discuss the simplicity of Aut (G). d) Explore the action of Aut(G) on the set of Sylow subgroups of G. Determine whether this action is transitive and justify your answer.

Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter4: More On Groups
Section4.4: Cosets Of A Subgroup
Problem 24E: Find two groups of order 6 that are not isomorphic.
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Let G be a finite group of order 56.
a) Determine the possible number of Sylow 7-subgroups in G. Provide a detailed justification
using Sylow's theorems.
b) Assume that G has exactly one Sylow 7-subgroup. Prove that this Sylow 7-subgroup is normal
in G, and subsequently show that G contains a normal subgroup of order 8.
c) Using the results from parts (a) and (b), classify all possible group structures for G. Discuss
whether G is necessarily abelian or non-abelian based on your classification.
Transcribed Image Text:Let G be a finite group of order 56. a) Determine the possible number of Sylow 7-subgroups in G. Provide a detailed justification using Sylow's theorems. b) Assume that G has exactly one Sylow 7-subgroup. Prove that this Sylow 7-subgroup is normal in G, and subsequently show that G contains a normal subgroup of order 8. c) Using the results from parts (a) and (b), classify all possible group structures for G. Discuss whether G is necessarily abelian or non-abelian based on your classification.
Let G be a finite non-abelian simple group of order 60.
a) Determine all possible isomorphism types of G. Justify your conclusion using the classification
of simple groups of small order.
b) Compute the automorphism group Aut(G) of G. Provide a detailed analysis of its structure,
including any inner and outer automorphisms.
c) Prove that every automorphism of G is inner or describe the existence of outer automorphisms.
Use this to discuss the simplicity of Aut (G).
d) Explore the action of Aut(G) on the set of Sylow subgroups of G. Determine whether this
action is transitive and justify your answer.
Transcribed Image Text:Let G be a finite non-abelian simple group of order 60. a) Determine all possible isomorphism types of G. Justify your conclusion using the classification of simple groups of small order. b) Compute the automorphism group Aut(G) of G. Provide a detailed analysis of its structure, including any inner and outer automorphisms. c) Prove that every automorphism of G is inner or describe the existence of outer automorphisms. Use this to discuss the simplicity of Aut (G). d) Explore the action of Aut(G) on the set of Sylow subgroups of G. Determine whether this action is transitive and justify your answer.
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