# Pseudoverbal subgroup

## Contents

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
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]

## Definition

Suppose $\mathcal{V}$ is a subpseudovariety of the variety of groups, i.e., $\mathcal{V}$ is a collection of groups closed under taking subgroups, quotients, and direct products. Equivalently, the group property of being in $\mathcal{V}$ is a pseudovarietal group property.

The $\mathcal{V}$-pseudoverbal subgroup of a group $G$ is defined as the intersection of all normal subgroups $N$ of $G$ for which the quotient group $G/N$ is in $\mathcal{V}$. Note that the quotient group of $G$ by its $\mathcal{V}$-pseudoverbal subgroup need not itself be in the pseudovariety.

## Relation with other properties

### Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
verbal subgroup similar definition, but for a subvariety instead of a subpseudovariety |FULL LIST, MORE INFO

### Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
quotient-subisomorph-containing subgroup contained in the kernel of any homomorphism to the quotient group |FULL LIST, MORE INFO
fully invariant subgroup (via quotient-subisomorph-containing) (via quotient-subisomorph-containing) Quotient-subisomorph-containing subgroup|FULL LIST, MORE INFO
characteristic subgroup (via fully invariant) (via fully invariant) Fully invariant subgroup, Quotient-subisomorph-containing subgroup|FULL LIST, MORE INFO