# Quasiverbal subgroup

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## 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 subquasivariety of the variety of groups. Equivalently, the group property of being in $\mathcal{V}$ is a quasivarietal group property.

The $\mathcal{V}$-quasiverbal subgroup of a group $G$ is defined in the following equivalent ways:

• It is the intersection of all normal subgroups $N$ of $G$ for which the quotient group $G/N$ is in $\mathcal{V}$.
• It is the unique smallest normal subgroup of $G$ for which the quotient group is in the quasivariety $\mathcal{V}$.

## Metaproperties

Metaproperty name Satisfied? Proof Statement with symbols
quotient-transitive subgroup property Yes quasiverbality is quotient-transitive Suppose $H \le K \le G$ are groups such that $H$ is a quasiverbal subgroup of $G$ and $K/H$ is a quasiverbal subgroup of $G/H$. Then, $K$ is a quasiverbal subgroup of $G$.
strongly join-closed subgroup property Yes quasiverbality is strongly join-closed Suppose $H_i, i \in I$ are all quasiverbal subgroups of a group $G$. Then, the join of subgroups $\langle H_i \rangle_{i \in I}$ is also a quasiverbal subgroup.

## 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 subquasivariety verbal implies quasiverbal quasiverbal not implies verbal |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 invariant under all endomorphisms [(via quotient-subisomorph-containing) (via quotient-subisomorph-containing) Quotient-subisomorph-containing subgroup|FULL LIST, MORE INFO
characteristic subgroup invariant under all automorphisms (via fully invariant) (via fully invariant) Quotient-subisomorph-containing subgroup|FULL LIST, MORE INFO