# Characteristic not implies fully invariant

This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., characteristic subgroup) neednotsatisfy the second subgroup property (i.e., fully invariant subgroup)

View a complete list of subgroup property non-implications | View a complete list of subgroup property implications

Get more facts about characteristic subgroup|Get more facts about fully invariant subgroup

EXPLORE EXAMPLES YOURSELF: View examples of subgroups satisfying property characteristic subgroup but not fully invariant subgroup|View examples of subgroups satisfying property characteristic subgroup and fully invariant subgroup

## Statement

A characteristic subgroup of a group need not be a fully invariant subgroup.

## Definitions used

Term | Definition |
---|---|

characteristic subgroup | A subgroup of a group is termed a characteristic subgroup if for any automorphism of and any , . |

fully invariant subgroup | A subgroup of a group is termed a fully invariant subgroup if for any endomorphism of and any , . |

## Related facts

### For particular kinds of groups

- Every nontrivial characteristic subgroup is potentially characteristic-and-not-fully invariant
- Every nontrivial normal subgroup is potentially characteristic-and-not-fully invariant

### Some kinds of characteristic subgroups that are not fully invariant

- Center not is fully invariant
- Characteristic direct factor not implies fully invariant
- Characteristic not implies fully invariant in finite abelian group
- Characteristic not implies fully invariant in odd-order class two p-group
- Characteristic not implies fully invariant in class three maximal class p-group

### Opposite facts

### Some subgroup-defining functions yield fully invariant subgroups

All members of the derived series and the lower central series of a group are fully characteristic. This follows from the fact that they are all verbal subgroups (i.e., can be described as being generated by words of a certain form. `Further information: Verbal subgroup, Verbal implies fully characteristic`

## Proof

### The example of the dihedral group

`Further information: Dihedral group:D8`

In the dihedral group of order eight, there exists a cyclic characteristic subgroup of order four that is *not* fully characteristic. Specifically, if we have:

then the subgroup generated by is characteristic. On the other hand, it is *not* fully characteristic: consider the homomorphism that sends both and to . This is a retraction to the two-element subgroup , and under this retraction, the subgroup generated by is *not* invariant (since gets mapped to ).

### The example of the center, or a characteristic direct factor

If is a nontrivial centerless group and is an Abelian group isomorphic to a nontrivial Abelian subgroup of , then we have:

- is characteristic in : In fact, is the center of .
- is not fully characteristic in : Consider the endomorphism with kernel , mapping isomorphically to . is not invariant under this endomorphism.