# Odd-order abelian group not is fully invariant in holomorph

From Groupprops

## Statement

It is possible to have an Odd-order abelian group (?) such that is not a Fully invariant subgroup (?) in its holomorph.

The smallest counterexample is , i.e., the direct product of the cyclic groups of order and .

## Related facts

- 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
- Cyclic group not is characteristic in holomorph, cyclic group not is fully invariant in holomorph
- Additive group of a field implies characteristic in holomorph
- Odd-order elementary abelian group is fully invariant in holomorph