Burnside's theorem on coprime automorphisms and Frattini subgroup

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Statement

Suppose P is a finite p-group, i.e., a group of prime power order. Suppose \varphi is an automorphism of P whose order (as an element of the automorphism group of P) is relatively prime to P. Then, if \varphi induces the identity on P/\Phi(P), \varphi is the identity automorphism of P. Here \Phi(P) is the Frattini subgroup of P.

Equivalently the kernel of the map:

\operatorname{Aut}(P) \to \operatorname{Aut}(P/\Phi(P))

is a p-group. We also say that the Frattini subgroup of a p-group is a quotient-coprime automorphism-faithful subgroup.

Related facts

Generalizations

Similar facts

Facts used

  1. Frattini subgroup of finite group is quotient-coprime automorphism-faithful

Proof

Given: A finite p-group P, an automorphism \varphi of P whose order is relatively prime to p. Further \varphi induces the identity automorphism on P/\Phi(P).

To prove: \varphi is the identity automorphism.

Proof: The proof follows directly from fact (1), which is actually the general formulation for finite groups.

(NOTE: We can use the Burnside's basis theorem instead; instead of picking a coset representative for every coset, we pick a Burnside basis, and then find a coset representative for each coset in the Burnside basis, that is fixed under \sigma.)

References

Textbook references

  • Finite Groups by Daniel Gorenstein, ISBN 0821843427, More info, Page 199, Theorem 1.4 (Chapter 5)
  • Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347, More info, Page 199, Exercise 26(e) and (f) (Section 6.1)