Automorph-conjugacy is transitive
This article gives the statement, and possibly proof, of a subgroup property (i.e., automorph-conjugate subgroup) satisfying a subgroup metaproperty (i.e., transitive subgroup property)
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Suppose are groups such that is an automorph-conjugate subgroup of , and is an automorph-conjugate subgroup of . Then, is an automorph-conjugate subgroup of .
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Given: Groups such that is an automorph-conjugate subgroup of and is an automorph-conjugate subgroup of . An automorphism of .
To prove: There exists such that
|Step no.||Assertion/construction||Given data used||Previous steps used||Explanation|
|2||There exists such that|| is an automorph-conjugate subgroup of
is an automorphism of
|3||Denote by the map . Then is an automorphism of that restricts to an automorphism of .||Step (2)||direct from the step|
|4||There exists such that .||is an automorph-conjugate subgroup of||Step (3)||Step-given combination direct|
|5||Setting , we get that||Step (4)|| Simple algebraic manipulation gives |