# Isomorph-free subgroup

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

## Definition

QUICK PHRASES: no other isomorphic subgroups, no isomorphic copies, only subgroup of its isomorphism type

A subgroup of a group is said to be **isomorph-free** if it satisfies the following equivalent conditions:

- Whenever such that , then (i.e. and are the
*same*subgroup). - is a co-Hopfian group, and whenever such that , then .

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]

This is a variation of characteristic subgroup|Find other variations of characteristic subgroup | Read a survey article on varying characteristic subgroup

This article describes a property that arises as the conjunction of a subgroup property: isomorph-containing subgroup with a group property (itself viewed as a subgroup property): co-Hopfian group

View a complete list of such conjunctions

## Examples

### Extreme examples

- The trivial subgroup is isomorph-free.
- Any co-Hopfian group (and in particular, any finite group) is isomorph-free as a subgroup of itself.

### Examples in small finite groups

Below are some examples of a proper nontrivial subgroup that satisfy the property isomorph-free subgroup.

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

## Relation with other properties

### Stronger properties

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

normal Sylow subgroup | Sylow subgroup that is also normal | Follows from Sylow implies order-conjugate | Finite isomorph-free subgroup, Order-unique subgroup|FULL LIST, MORE INFO | |

normal Hall subgroup | Hall subgroup (i.e., order and index are relatively prime) that is also normal | Finite isomorph-free subgroup, Order-unique subgroup|FULL LIST, MORE INFO | ||

order-unique subgroup | unique subgroup of its order | isomorph-free not implies order-unique (see also list of examples) | |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-freeness is not transitive | It is possible to have groups such that is isomorph-free in and is isomorph-free in but is not isomorph-free in . | |

transitive subgroup property | Yes | isomorph-freeness is quotient-transitive | Suppose are groups such that is isomorph-free in and the quotient group is isomorph-free in . Then, is isomorph-free in . | |

intermediate subgroup condition | Yes | isomorph-freeness satisfies intermediate subgroup condition | Suppose are groups such that is isomorph-free in . Then, is also isomorph-free in . | |

finite-intersection-closed subgroup property | No | isomorph-freeness is not finite-intersection-closed | It is possible to have a group and subgroups of such that and are both isomorph-free but the intersection is not isomorph-free. | |

strongly join-closed subgroup property | Yes | isomorph-freeness is strongly join-closed | Suppose are subgroups of a group such that each is an isomorph-free subgroup of . Then, the join of subgroups is also an isomorph-free subgroup of . | |

finite-upper join-closed subgroup property | No | isomorph-freeness is not finite-upper join-closed | It is possible to have groups and are intermediate subgroups such that is isomorph-free in both and , but is not isomorph-free in . | |

trivially true subgroup property | Yes | The trivial subgroup is isomorph-free in any group. | ||

identity-true subgroup property | No | It is possible for a group to be isomorphic to a subgroup of itself. The simplest example is , the group of integers, that is isomorphic to the subgroup for any positive integer . A group that is isomorph-free as a subgroup of itself is termed a co-Hopdian group. |

## Effect of property operators

### The subordination operator

*Applying the subordination operator to this property gives*: sub-isomorph-free subgroup

A subgroup of a group is termed **sub-isomorph-free** if there is a series of subgroups , with each an isomorph-free subgroup of .

## Testing

### GAP code

GAP-codable subgroup propertyOne can write code to test this subgroup property inGAP (Groups, Algorithms and Programming), though there is no direct command for it.

View the GAP code for testing this subgroup property at: IsIsomorphFreeSubgroup

View other GAP-codable subgroup properties | View subgroup properties with in-built commands

While there is no in-built command for testing whether a subgroup is isomorph-free, a short piece of GAP code can do the test. The code can be found at GAP:IsIsomorphFreeSubgroup, and the command is invoked as follows:

IsIsomorphFreeSubgroup(group,subgroup);