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]

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