Multi-invariance property

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

Definition

A subgroup property is termed a multi-invariance property if there exists, for every natural number $n$, a collection of functions $F_n$ from $G^n$ to $G$, such that a subgroup $H$ of $G$ satisfies the subgroup property iff the following holds:

$\mathrm{\forall }n,\mathrm{\forall }f\in F_n,f(h_1,h_2,\dots ,h_n)\in H\mathrm{\forall }{h}_1,h_2,\dots ,h_n\in H$

A special case of this is an invariance property where the $F_n$ are empty for $n>1$.

Relation with other metaproperties

Stronger metaproperties

• Invariance property

Weaker metaproperties

• Strongly intersection-closed subgroup property: For full proof, refer: Multi-invariance implies strongly intersection-closed
• Identity-true subgroup property
• Intersection-closed subgroup property
• ACU-closed subgroup property: For full proof, refer: Multi-invariance implies ACU-closed

Facts

• Multi-invariance equals strongly intersection-closed in Noetherian group