# Difference between revisions of "Isomorph-containing subgroup"

From Groupprops

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==Metaproperties== | ==Metaproperties== | ||

− | {{ | + | {{wikilocal-section}} |

− | + | Here is a summary: | |

− | {{ | + | {| class="sortable" border="1" |

− | + | !Metaproperty name !! Satisfied? !! Proof !! Difficulty level (0-5) !! Statement with symbols | |

− | { | + | |- |

+ | |[[satisfies metaproperty::transitive subgroup property]] || No || [[isomorph-containment is not transitive]] || {{#show: isomorph-containment is not transitive | ?Difficulty level}} || It is possible to have groups <math>H \le K \le G</math> such that <math>H</math> is an isomorph-containing subgroup of <math>K</math> and <math>K</math> is an isomorph-containing subgroup of <math>G</math>, but <math>H</math> is not an isomorph-containing subgroup of <math>G</math>. | ||

+ | |- | ||

+ | |[[satisfies metaproperty::trim subgroup property]] || Yes || || 0 || In any group <math>G</math>, the trivial subgroup <math>\{ e \}</math> and the whole group <math>G</math> are both isomorph-containing subgroups of <math>G</math> | ||

+ | |- | ||

+ | |[[satisfies metaproperty::intermediate subgroup condition]] || Yes || || 1 || If <math>H \le K \le G</math> are groups such that <math>H</math> is isomorph-containing in <math>G</math>, then <math>H</math> is also isomorph-containing in <math>K</math> | ||

+ | |} |

## Revision as of 00:36, 18 March 2019

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

QUICK PHRASES: contains all isomorphic subgroups, weakly closed in any ambient group

A subgroup of a group is termed an **isomorph-containing subgroup** if it satisfies the following equivalent conditions:

- Whenever is a subgroup of isomorphic to , .
- If is a subgroup of , is weakly closed in with respect to .

### Equivalence of definitions

`Further information: Isomorph-containing iff weakly closed in any ambient group`

## Examples

VIEW: subgroups of groups satisfying this property | subgroups of groups dissatisfying this propertyVIEW: Related subgroup property satisfactions | Related subgroup property dissatisfactions

## Relation with other properties

### Stronger properties

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

isomorph-free subgroup | no other isomorphic subgroup (for a finite subgroup, and more generally, for a co-Hopfian subgroup, the two properties are equivalent) | obvious | any example of a non-co-Hopfian group as a subgroup of itself -- such as the group of integers | |

homomorph-containing subgroup | contains any homomorphic image of itself in the group | obvious; isomorphs are also homomorphic images | cyclic maximal subgroup of dihedral group:D8 is isomorph-containing but not homomorph-containing | |FULL LIST, MORE INFO |

Subhomomorph-containing subgroup | contains any homomorphic image in the whole group of any subgroup of it | (via homomorph-containing) | (via homomorph-containing) | Homomorph-containing subgroup, Right-transitively isomorph-containing subgroup, Subisomorph-containing subgroup|FULL LIST, MORE INFO |

subisomorph-containing subgroup | contains any subgroup of the whole group isomorphic to any subgroup of itself | (obvious) | cyclic maximal subgroup of dihedral group:D8 is isomorph-containing but not subisomorph-containing | Right-transitively isomorph-containing subgroup|FULL LIST, MORE INFO |

variety-containing subgroup | contains any subgroup of the whole group in the subvariety of the variety of groups generated by it | (obvious) | (via either homomorph-containing or subisomorph-containing) | Homomorph-containing subgroup, Right-transitively isomorph-containing subgroup, Subisomorph-containing subgroup|FULL LIST, MORE INFO |

fully invariant direct factor | both a fully invariant subgroup and a direct factor | equivalence of definitions of fully invariant direct factor | Complemented homomorph-containing subgroup, Complemented isomorph-containing subgroup, Homomorph-containing subgroup|FULL LIST, MORE INFO |

### Weaker properties

## 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 | Difficulty level (0-5) | Statement with symbols |
---|---|---|---|---|

transitive subgroup property | No | isomorph-containment is not transitive | It is possible to have groups such that is an isomorph-containing subgroup of and is an isomorph-containing subgroup of , but is not an isomorph-containing subgroup of . | |

trim subgroup property | Yes | 0 | In any group , the trivial subgroup and the whole group are both isomorph-containing subgroups of | |

intermediate subgroup condition | Yes | 1 | If are groups such that is isomorph-containing in , then is also isomorph-containing in |