Bryant-Kovacs theorem
From Groupprops
Statement
Suppose is a prime number, and is a vector space over the prime field of dimension greater than . In other words, is an elementary Abelian -group that is not cyclic.
Then, if is a subgroup of (the general linear group on ), there exists a finite -group such that , and under the natural homomorphism:
the image of is precisely .
Related facts
- Burnside's theorem on coprime automorphisms and Frattini subgroup: This states that the kernel of the natural homomorphism from to is a -group.
Corollaries
- Every p'-group is the p'-part of the automorphism group of a p-group
- Every elementary abelian p-group occurs as the Frattini quotient of a p-group in which every maximal subgroup is characteristic
- Finite elementary abelian implies single-witness FQPAC
References
Textbook references
- Finite Groups II by Bertram Huppert and Norman Blackburn, ISBN 0387106324 (English), ISBN 3540106324 (German), Page 403, Theorem 13.5, Chapter 13 (Automorphisms of p-groups), ^{More info}