# 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
View a complete list of subgroup metaproperties
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## 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: