Intersection of finitely many verbal subgroups

This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

|Get subgroup property lookup help |Get exploration suggestions
VIEW RELATED: Subgroup property implications | Subgroup property non-implications | Subgroup metaproperty satisfactions | Subgroup metaproperty dissatisfactions | Subgroup property satisfactions |Subgroup property dissatisfactions
VIEW FACTS THAT:use, or start with, a subgroup satisfying the property|prove, or show that, a subgroup satisfies the property

Definition

Suppose $G$ is a group and $H$ is a subgroup of $G$. We say that $H$ is an intersection of finitely many verbal subgroups in $G$ if there exists a positive integer $n$ and verbal subgroups $H_{1},H_{2},\dots ,H_{n}$ of $G$ such that $H$ equals the intersection of subgroups $\bigcap _{i=1}^{n}H_{i}$.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
verbal subgroup	generated by a set of words			|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
existentially bound-word subgroup				|FULL LIST, MORE INFO
fully invariant subgroup	invariant under all endomorphisms			|FULL LIST, MORE INFO
characteristic subgroup	invariant under all automorphisms			Fully invariant subgroup|FULL LIST, MORE INFO
normal subgroup	invariant under all inner automorphisms			Fully invariant subgroup|FULL LIST, MORE INFO