# Join of homomorph-dominating subgroups

From Groupprops

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]

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## Definition

A subgroup of a group is termed a **join of homomorph-dominating subgroups** if there exists a collection of subgroups of such that is the join .

## Relation with other properties

### Stronger properties

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

homomorph-dominating subgroup | every homomorphic image is contained in a conjugate | (obvious) | homomorph-domination is not finite-join-closed | |FULL LIST, MORE INFO |

homomorph-containing subgroup | contains every homomorphic image | (via homomorph-dominating) | |FULL LIST, MORE INFO | |

join of Sylow subgroups | join of Sylow subgroups (with no restriction on whether the primes need be the same or different) of the whole group | follows from Sylow implies homomorph-dominating | |FULL LIST, MORE INFO | |

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

## Formalisms

### In terms of the join-closure operator

This property is obtained by applying the join-closure operator to the property: homomorph-dominating subgroup

View other properties obtained by applying the join-closure operator