# Right-transitively homomorph-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

A subgroup of a group is termed a **right-transitively homomorph-containing subgroup** if, whenever is a homomorph-containing subgroup of , is also a homomorph-containing subgroup of .

## Relation with other properties

### Stronger properties

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

subhomomorph-containing subgroup | contains every homomorphic image of every subgroup | subhomomorph-containing implies right-transitively homomorph-containing | right-transitively homomorph-containing not implies subhomomorph-containing | |FULL LIST, MORE INFO |

order-containing subgroup | contains every subgroup whose order divides its order | (via subhomomorph-containing) | (via subhomomorph-containing) | |FULL LIST, MORE INFO |

variety-containing subgroup | contains every subgroup of the whole group in the variety it generates | (via subhomomorph-containing) | (via subhomomorph-containing) | |FULL LIST, MORE INFO |

### Weaker properties

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

homomorph-containing subgroup | contains every homomorphic image of itself | (obvious) | homomorph-containment is not transitive | |FULL LIST, MORE INFO |

fully invariant subgroup | contains every image of itself under an endomorphism of the whole group | (via homomorph-containing) | (via homomorph-containing) | Homomorph-containing subgroup|FULL LIST, MORE INFO |

characteristic subgroup | contains every image of itself under an automorphism of the whole group | (via fully invariant) | (via fully invariant) | Homomorph-containing subgroup|FULL LIST, MORE INFO |

normal subgroup | contains every image of itself under an inner automorphism of the whole group | (via characteristic) | (via characteristic) | Homomorph-containing subgroup|FULL LIST, MORE INFO |