Odd-order cyclic group is fully invariant in holomorph

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Statement

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

Related facts

Facts used

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

Proof

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