Commutator of finite group with cyclic coprime automorphism group equals second commutator

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Let G be a finite group and A be a cyclic subgroup of \operatorname{Aut}(G) such that the orders of G and A are relatively prime. Then:

[[G,A],A] = [G,A].

Facts used

  1. Centralizer-commutator product decomposition for finite groups
  2. Cyclicity is quotient-closed
  3. Lagrange's theorem
  4. Order of quotient group divides order of group


Textbook references