Quasiverbal subgroup

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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 $H\leq K\leq G$ 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)	|FULL LIST, MORE INFO
characteristic subgroup	invariant under all automorphisms	(via fully invariant)	(via fully invariant)	|FULL LIST, MORE INFO