ACIC is characteristic subgroup-closed
This article gives the statement, and possibly proof, of a group property (i.e., ACIC-group) satisfying a group metaproperty (i.e., characteristic subgroup-closed group property)
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Suppose is an ACIC-group, i.e., every automorph-conjugate subgroup of is characteristic in (note that characteristic implies automorph-conjugate, so in such a group, the notion of being a characteristic subgroup precisely coincides with the notion of being an automorph-conjugate subgroup).
Suppose is a characteristic subgroup (or equivalently, an automorph-conjugate subgroup) of . Then, is also an ACIC-group.
|automorph-conjugate subgroup||A subgroup of a group is termed automorph-conjugate if any subgroup of automorphic to is conjugate to .|
|ACIC-group||A group is termed ACIC if every automorph-conjugate subgroup of is characteristic in ; equivalently, every automorph-conjugate subgroup of is normal in .|
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Given: An ACIC-group , a characteristic subgroup (or equivalently, an automorph-conjugate subgroup) of , and an automorph-conjugate subgroup of .
To prove: is normal in , or equivalently, is characteristic in .
|Step no.||Assertion/construction||Facts used||Given data used||Previous steps used||Explanation|
|1||is automorph-conjugate in .||Fact (1)||is automorph-conjugate in , is automorph-conjugate in .||Fact-given direct.|
|2||is normal in .||is ACIC.||Step (1)||Step-given direct.|
|3||is normal in .||Fact (2)||.||Step (2)||Step-given-fact direct.|