Automorphism-based relation-implication-expressible subgroup property
This article defines a subgroup metaproperty: a property that can be evaluated to true/false for any subgroup property
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Suppose and are two group-closed properties of automorphisms from a group to itself. Let denote the following relation between subgroups: two subgroups satisfy if there is an automorphism of satisfying property and sending to . Analogously, define . The automorphism-based relation-implication-expressible subgroup property for and is the property with relation implication expression:
In other words, satisfies property in if whenever satisfy (i.e., there is an automorphism with property sending to ), also satisfy property (i.e., there is an automorphism with property sending to ).
Here are some examples:
- Characteristic subgroup: Here, the two properties are automorphism and identity map.
- Normal subgroup: Here, the two properties are inner automorphism and identity map.
- Automorph-conjugate subgroup: Here, the two properties are automorphism and inner automorphism.