Pseudoverbal subgroup

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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]

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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)	|FULL LIST, MORE INFO
characteristic subgroup		(via fully invariant)	(via fully invariant)	Fully invariant subgroup|FULL LIST, MORE INFO