Conjugate-intersection-closed 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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Definition

Symbol-free definition

A subgroup property is termed conjugate-intersection-closed if whenever a subgroup has the property, then any intersection of a family of conjugate subgroups of that subgroup also has that property.

Definition with symbols

We say that property $p$ is conjugate-intersection-closed if the following holds.

Suppose $G$ is a group, $H$ is a subgroup, and $S$ is any subset of $G$. Then, if $H$ has property $p$ in $G$ so does the group:

${\bigcap }_{g\in S}gH{g}^{-1}$

Relation with other metaproperties

Stronger metaproperties

• Intersection-closed subgroup property
• Automorph-intersection-closed subgroup property

Weaker metaproperties

• Normal core-closed subgroup property