Bryant-Kovacs theorem

Statement
Suppose $$p$$ is a prime number, and $$V$$ is a vector space over the prime field $$\mathbb{F}_p$$ of dimension greater than $$1$$. In other words, $$V$$ is an elementary Abelian $$p$$-group that is not cyclic.

Then, if $$G$$ is a subgroup of $$GL(V)$$ (the general linear group on $$V$$), there exists a finite $$p$$-group $$P$$ such that $$P/\Phi(P) \cong V$$, and under the natural homomorphism:

$$\operatorname{Aut}(P) \to \operatorname{Aut}(P/\Phi(P)) = GL(V)$$

the image of $$\operatorname{Aut}(P)$$ is precisely $$G$$.

Related facts

 * Burnside's theorem on coprime automorphisms and Frattini subgroup: This states that the kernel of the natural homomorphism from $$\operatorname{Aut}(P)$$ to $$\operatorname{Aut}(P/\Phi(P))$$ is a $$p$$-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