Automorph-conjugacy is normalizer-closed
This article gives the statement, and possibly proof, of a subgroup property (i.e., automorph-conjugate subgroup) satisfying a subgroup metaproperty (i.e., normalizer-closed subgroup property)
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- Automorphism-based relation-implication-expressible implies normalizer-closed: This is a general formalism result stating that any subgroup property that can be expressed in a given format is closed under taking normalizers.
The key idea behind the proof is that taking the normalizer commutes with automorphisms. In other words, for any automorphism , we have .
Given: A group , an automorph-conjugate subgroup .
To prove: is an automorph-conjugate subgroup: given any automorphism of , and are conjugate.
Proof: . By assumption, is automorph-conjugate, so there exists a such that . Thus, , and thus, and are conjugate by .