# Odd-order cyclic group is fully invariant in holomorph

From Groupprops

## Contents

## Statement

Any Odd-order cyclic group (?) is a Fully invariant subgroup (?) inside its holomorph.

## Related facts

- Odd-order abelian group not is fully invariant in holomorph: The analogous statement is not true for odd-order abelian groups.
- Cyclic group not is fully invariant in holomorph: The analogous statement is not true if we remove the conditions of odd order. In fact, if divides the order of a cyclic group, then it is not fully characteristic in its holomorph.
- Cyclic group not is characteristic in holomorph

## Facts used

- Odd-order cyclic group equals commutator subgroup of holomorph
- Commutator subgroup is fully characteristic

## Proof

The proof follows directly from facts (1) and (2).