Groupprops, The Group Properties Wiki (pre-alpha)

From Groupprops

Contents 1 Statement 2 Related facts 2.1 Corollaries 3 References 3.1 Textbook references

Statement

Suppose p is a prime number, and V is a vector space over the prime field 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 , and under the natural homomorphism:

the image of is precisely G.

Related facts

• Burnside's theorem on coprime automorphisms and Frattini subgroup: This states that the kernel of the natural homomorphism from to 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}