Automorphism-based relation-implication-expressible subgroup property

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This article defines a subgroup metaproperty: a property that can be evaluated to true/false for any subgroup property
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VIEW RELATED: subgroup metaproperty satisfactions| subgroup metaproperty dissatisfactions

Statement

Suppose a and b are two group-closed properties of automorphisms from a group to itself. Let R_a denote the following relation between subgroups: two subgroups H, K \le G satisfy R_a if there is an automorphism of G satisfying property a and sending H to K. Analogously, define R_b. The automorphism-based relation-implication-expressible subgroup property for a and b is the property with relation implication expression:

p = R_a \implies R_b.

In other words, H satisfies property p in G if whenever (H,K) satisfy R_a (i.e., there is an automorphism with property a sending H to K), (H,K) also satisfy property R_b (i.e., there is an automorphism with property b sending H to K).

Examples

Here are some examples:

Relation with other metaproperties

Stronger metaproperties

Weaker metaproperties