# Quasiverbal 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 subquasivariety of the variety of groups. Equivalently, the group property of being in is a quasivarietal group property.

The -quasiverbal subgroup of a group is defined in the following equivalent ways:

- It is the intersection of all normal subgroups of for which the quotient group is in .
- It is the unique smallest normal subgroup of for which the quotient group is in the quasivariety .

## Metaproperties

Metaproperty name | Satisfied? | Proof | Statement with symbols |
---|---|---|---|

quotient-transitive subgroup property | Yes | quasiverbality is quotient-transitive | Suppose are groups such that is a quasiverbal subgroup of and is a quasiverbal subgroup of . Then, is a quasiverbal subgroup of . |

strongly join-closed subgroup property | Yes | quasiverbality is strongly join-closed | Suppose are all quasiverbal subgroups of a group . Then, the join of subgroups 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 |