# Cyclic group not is characteristic in holomorph

From Groupprops

This article gives the statement, and possibly proof, of a particular subgroup or type of subgroupnotsatisfying a particular subgroup property (namely, Characteristic subgroup (?)) in a particular group or type of group .

## Statement

A cyclic group need *not* be a characteristic subgroup inside its holomorph. In fact, if divides the order of the cyclic group, it is not characteristic inside its holomorph.

## Related facts

- Cyclic group not is fully invariant in holomorph: This breaks down even when divides the order of the group.
- Odd-order cyclic group equals commutator subgroup of holomorph
- Odd-order cyclic group is fully invariant in holomorph
- Odd-order cyclic group is characteristic in holomorph
- Odd-order elementary abelian group is fully invariant in holomorph
- Odd-order abelian group not is fully invariant in holomorph