Sylow implies automorphconjugate
From Groupprops
This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property must also satisfy the second subgroup property
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This fact is an application of the following pivotal fact/result/idea: Sylow's theorem
View other applications of Sylow's theorem OR Read a survey article on applying Sylow's theorem
Contents
Statement
In a finite group, any Sylow subgroup is an automorphconjugate subgroup.
Definitions used
Sylow subgroup
Further information: Sylow subgroup
Automorphconjugate subgroup
Further information: Automorphconjugate subgroup
Proof
Handson proof
Given: A finite group , a Sylow subgroup
To prove: For any automorphism of , and are conjugate
Proof: The key thing to observe is that is also a Sylow subgroup. Hence, and are Sylow subgroups, so by the conjugacy part of Sylow's theorem, they are conjugate.