# Autoclinism-invariant subgroup

From Groupprops

## Definition

### As an invariance property

A subgroup of a group is termed an **autoclinism-invariant subgroup** if it is invariant under any bijective set map satisfying all three of these conditions:

- induces an automorphism modulo the center, i.e., mod .
- restricts to an automorphism on the derived subgroup, i.e., sends the derived subgroup to itself and the restriction to the derived subgroup is an automorphism of the derived subgroup.
- for all (possibly equal, possibly distinct) .

### As a two-case property

A subgroup of a group is termed an **autoclinism-invariant subgroup** if it satisfies *either* of these conditions:

- contains the center and is invariant under any automorphism of that is the inner automorphism group part of the data specifying an autoclinism.
- is contained in the derived subgroup and it is invariant under any automorphism of that is the derived subgroup part of the data specifying an autoclinism.

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]

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

## Examples

- Every group is autoclinism-invariant in itself.
- The trivial subgroup is autoclinism-invariant in any group.
- All members of the upper central series and all members of the lower central series are autoclinism-invariant.
- For finite p-groups, the ZJ-subgroup and D*-subgroup are autoclinism-invariant.

## Relation with other properties

### Stronger properties

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

endoclinism-invariant subgroup | |FULL LIST, MORE INFO |

### Weaker properties

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

characteristic subgroup | invariant under all automorphisms | |FULL LIST, MORE INFO | ||

normal subgroup | invariant under all inner automorphisms | |FULL LIST, MORE INFO |