# Homomorph-containing subgroup

From Groupprops

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

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

### Equivalent definitions in tabular format

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

1 | contains every homomorphic image | it contains any homomorphic image of itself in the whole group. | for any homomorphism of groups , . |

2 | homomorphism to whole group restricts to endomorphism | every homomorphism of groups from the subgroup to the whole group restricts to an endomorphism of the suubgrop. | for any homomorphism of groups , and the restriction of to is an endomorphism of . |

3 | (definition in terms of Hom-set maps) | (too complicated to state without symbols) | the natural map (by inclusion) is a surjective map of sets. |

## Examples

### Extreme examples

- Every group is homomorph-containing as a subgroup of itself.
- The trivial subgroup is homomorph-containing in any group.

### Important classes of examples

- Normal Sylow subgroups and normal Hall subgroups are homomorph-containing.
- Subgroups defined as the subgroup generated by elements of specific orders, are all homomorph-containing subgroups. The omega subgroups of a group of prime power order are such examples.
`Further information: Omega subgroups are homomorph-containing` - The perfect core of a group is a homomorph-containing subgroup.

See also the section #Stronger properties in this page.

### Examples in small finite groups

Below are some examples of a proper nontrivial subgroup that satisfy the property homomorph-containing subgroup.

Below are some examples of a proper nontrivial subgroup that *does not* satisfy the property homomorph-containing subgroup.

## Metaproperties

BEWARE!This section of the article uses terminology local to the wiki, possibly without giving a full explanation of the terminology used (though efforts have been made to clarify terminology as much as possible within the particular context)

Here is a summary:

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

trim subgroup property | Yes | For any group , both (as a subgroup of itself) and the trivial subgroup of are homomorph-containing subgroups of . | |

transitive subgroup property | No | homomorph-containment is not transitive | It is possible to have groups such that is homomorph-containing in and is homomorph-containing in but is not homomorph-containing in . |

intermediate subgroup condition | Yes | homomorph-containment satisfies intermediate subgroup condition | If and is homomorph-containing in , then is homomorph-containing in . |

strongly join-closed subgroup property | Yes | homomorph-containment is strongly join-closed | If are a collection of homomorph-containing subgroups of , the join of subgroups is also a homomorph-containing subgroup. |

quotient-transitive subgroup property | Yes | homomorph-containment is quotient-transitive | If such that is homomorph-containing in and is homomorph-containing in , then is homomorph-containing in . |