# Difference between revisions of "Isomorph-containing subgroup"

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{{quick phrase|[[quick phrase::contains all isomorphic subgroups]], [[quick phrase::weakly closed in any ambient group]]}} | {{quick phrase|[[quick phrase::contains all isomorphic subgroups]], [[quick phrase::weakly closed in any ambient group]]}} | ||

− | + | ===Equivalent definitions in tabular format=== | |

− | + | Below are many '''equivalent''' definitions of characteristic subgroup. | |

− | + | ||

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

+ | ! No. !! Shorthand !! A [[subgroup]] of a [[group]] is isomorph-containing in it if... !! A subgroup <math>H</math> of a group <math>G</math> is called an isomorph-containing subgroup of <math>G</math> if ... | ||

+ | |- | ||

+ | | 1 || contains all isomorphic subgroups || it contains any subgroup of the whole group isomorphic to it || for any subgroup <math>M</math> of <math>G</math> such that <math>H</math> and <math>M</math> are isomorphic groups, <math>M \le H</math> | ||

+ | |- | ||

+ | | 2 || weakly closed in any ambient group || it is weakly close in any ambient group of the whole group || for any group <math>L</math> containing <math>G</math>, <math>H</math> is [[defining ingredient::weakly closed subgroup|weakly closed]] in <math>G</math> with respect to <math>L</math> | ||

+ | |} | ||

===Equivalence of definitions=== | ===Equivalence of definitions=== |

## Revision as of 00:39, 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

### Equivalent definitions in tabular format

Below are many **equivalent** definitions of characteristic subgroup.

No. | Shorthand | A subgroup of a group is isomorph-containing in it if... | A subgroup of a group is called an isomorph-containing subgroup of if ... |
---|---|---|---|

1 | contains all isomorphic subgroups | it contains any subgroup of the whole group isomorphic to it | for any subgroup of such that and are isomorphic groups, |

2 | weakly closed in any ambient group | it is weakly close in any ambient group of the whole group | for any group containing , 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 |