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

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}