Odd-order cyclic group is characteristic in holomorph

Statement
Any fact about::odd-order cyclic group is a fact about::characteristic subgroup inside its holomorph.

Related facts about odd-order cyclic groups

 * Odd-order cyclic group equals commutator subgroup of holomorph
 * Odd-order cyclic group is fully invariant in holomorph
 * Odd-order abelian group not is fully invariant in holomorph

Breakdown for other cyclic groups

 * Cyclic group not is characteristic in holomorph
 * Cyclic group not is fully invariant in holomorph

Other related facts

 * Additive group of a field implies characteristic in holomorph, because additive group of a field is monolith in holomorph (in particular, any elementary abelian group is characteristic in its holomorph).
 * Odd-order elementary abelian group is fully invariant in holomorph
 * Odd-order abelian group not is fully invariant in holomorph

Facts used

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

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