# Subgroup whose normal closure is homomorph-containing

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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]

BEWARE!This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

## Definition

A subgroup of a group is termed a **subgroup whose normal closure is homomorph-containing** if , its normal closure in the whole group , is a homomorph-containing subgroup of .

## Relation with other properties

### Stronger properties

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

homomorph-dominating subgroup | any subgroup of the whole group that is a homomorphic image of the subgroup is contained in one of its conjugates | Join of homomorph-dominating subgroups|FULL LIST, MORE INFO | ||

Sylow subgroup | subgroup of finite group with maximal prime power order | (via homomorph-dominating) | Join of Sylow subgroups|FULL LIST, MORE INFO | |

join of Sylow subgroups | join of Sylow subgroups | follows from statement for Sylow subgroups, and homomorph-containment is strongly join-closed | Join of homomorph-dominating subgroups|FULL LIST, MORE INFO | |

Hall subgroup | order and index are relatively prime | (via join of Sylow subgroups) | Join of Sylow subgroups, Join of homomorph-dominating subgroups|FULL LIST, MORE INFO |

### Weaker properties

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

subgroup whose normal closure is fully invariant | normal closure is a fully invariant subgroup | follows from homomorph-containing implies fully invariant | follows from fully invariant not implies homomorph-containing, and the fact that since fully invariant implies normal, a fully invariant subgroup is its own normal closure | |FULL LIST, MORE INFO |

subgroup whose normal closure is characteristic | normal closure is a characteristic subgroup | (via subgroup whose normal closure is fully invariant, using fully invariant implies characteristic) | (via subgroup whose normal closure is fully invariant) | Subgroup whose normal closure is fully invariant|FULL LIST, MORE INFO |