# Finite direct power-closed characteristic subgroup

From Groupprops

## Definition

Suppose is a subgroup of a group . We say that is **finite direct power-closed characteristic** in if the following holds for every natural number : in the group which is defined as the external direct product of copies of , the corresponding subgroup is a characteristic 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]

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

## Examples

### Extreme examples

- The trivial subgroup in any group is a finite direct power-closed characteristic subgroup.
- Every group is a finite direct power-closed characteristic subgroup of itself.

### Examples of subgroup-defining functions

- Center is finite direct power-closed characteristic: The center of a group always has this property. More generally, any bound-word subgroup, and in particular any marginal subgroup, is finite direct power-closed characteristic. Thus, all members of the finite part of the upper central series are finite direct power-closed characteristic.
- The derived subgroup, and more generally any verbal subgroup or even any fully invariant subgroup, is finite direct power-closed characteristic. Hence, all members of the finite part of the lower central series as well as of the derived series are finite direct power-closed characteristic.

### Examples in small finite groups

Below are some examples of a proper nontrivial subgroup that satisfy the property finite direct power-closed characteristic subgroup.

Below are some examples of a proper nontrivial subgroup that *does not* satisfy the property finite direct power-closed characteristic subgroup.

## Metaproperties

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

transitive subgroup property | Yes | finite direct power-closed characteristic is transitive | If , with a finite direct power-closed characteristic subgroup of and a finite direct power-closed characteristic subgroup of , then is a finite direct power-closed characteristic subgroup of . |

quotient-transitive subgroup property | Yes | finite direct power-closed characteristic is quotient-transitive | If , with a finite direct power-closed characteristic subgroup of and a finite direct power-closed characteristic subgroup of , then is a finite direct power-closed characteristic subgroup of . |

strongly intersection-closed subgroup property | Yes | finite direct power-closed characteristic is strongly intersection-closed | If are all finite direct power-closed characteristic subgroups of , so is the intersection . |

centralizer-closed subgroup property | Yes | finite direct power-closed characteristic is centralizer-closed | If is a finite direct power-closed characteristic subgroup, so is the centralizer . |

commutator-closed subgroup property | Yes | finite direct power-closed characteristic is commutator-closed | If are both finite direct power-closed characteristic, then so is the commutator . |

## Relation with other properties

### Stronger properties

### Weaker properties

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

characteristic subgroup | invariant under all automorphisms | (by definition) | characteristicity is not finite direct power-closed | |FULL LIST, MORE INFO |

normal subgroup | invariant under all inner automorphisms | (via characteristic) | (via characteristic) | Characteristic subgroup|FULL LIST, MORE INFO |