Burnside's theorem on coprime automorphisms and Frattini subgroup
Contents
Statement
Suppose is a finite -group, i.e., a group of prime power order. Suppose is an automorphism of whose order (as an element of the automorphism group of ) is relatively prime to . Then, if induces the identity on , is the identity automorphism of . Here is the Frattini subgroup of .
Equivalently the kernel of the map:
is a -group. We also say that the Frattini subgroup of a -group is a quotient-coprime automorphism-faithful subgroup.
Related facts
Generalizations
- Frattini subgroup of finite group is quotient-coprime automorphism-faithful: This generalizes the statement of Burnside's theorem to an arbitrary finite group.
Similar facts
- Stability group of subnormal series of p-group is p-group: The result is similar, and it uses a similar proof technique.
- Omega-1 of odd-order p-group is coprime automorphism-faithful
Facts used
Proof
Given: A finite -group , an automorphism of whose order is relatively prime to . Further induces the identity automorphism on .
To prove: 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 .)
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)