# Pseudoverbal subgroup

From Groupprops

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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 is a subpseudovariety of the variety of groups, i.e., is a collection of groups closed under taking subgroups, quotients, and direct products. Equivalently, the group property of being in is a pseudovarietal group property.

The -pseudoverbal subgroup of a group is defined as the intersection of all normal subgroups of for which the quotient group is in . Note that the quotient group of by its -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 |