Odd-order cyclic group is fully invariant in holomorph

Statement
Any fact about::odd-order cyclic group is a fact about::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 $$4$$ 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

 * 1) uses::Odd-order cyclic group equals commutator subgroup of holomorph
 * 2) uses::Commutator subgroup is fully characteristic

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