# Quotient-subisomorph-containing 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

The following are some equivalent definitions of **quotient-subisomorph-containing subgroup**.

No. | Shorthand | A subgroup of a group is termed a quasiverbal subgroup if ... |
---|---|---|

1 | contained in the kernel of any homomorphism to the quotient | for any homomorphism of groups , is contained in the kernel of . |

2 | intersection of normal subgroups whose quotients satisfy a subgroup-closed group property | there exists a subgroup-closed group property such that is the intersection of all normal subgroups of for which the quotient group satisfies property . |

3 | smallest normal subgroup whose quotient satisfies a subgroup-closed group property | there exists a subgroup-closed group property such that is the unique smallest normal subgroup of for which the quotient group satisfies . Note that every that works for the preceding definition need not work for this definition. |

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

verbal subgroup | union of the images of word maps | Pseudoverbal subgroup, Quasiverbal subgroup|FULL LIST, MORE INFO | ||

pseudoverbal subgroup | intersection of normal subgroups with quotients in a pseudovariety | |FULL LIST, MORE INFO | ||

quasiverbal subgroup | intersection of normal subgroups with quotients in a quasivariety | |FULL LIST, MORE INFO | ||

normal subgroup having no nontrivial homomorphism to its quotient group | no nontrivial homomorphism from the subgroup to the quotient group | |FULL LIST, MORE INFO | ||

normal Sylow subgroup | normal and a Sylow subgroup | Normal subgroup having no nontrivial homomorphism to its quotient group|FULL LIST, MORE INFO | ||

Normal Hall subgroup | normal and a Hall subgroup | quotient-subisomorph-containing subgroup|normal Hall subgroup}} |

### Weaker properties

- Weakly image-closed fully invariant subgroup
- Fully invariant subgroup
- Strictly characteristic subgroup
- Characteristic subgroup