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 .
- Burnside's theorem on coprime automorphisms and Frattini subgroup: This states that the kernel of the natural homomorphism from to is a -group.
- 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