Representations of elementary abelian p-groups and finite subgroups of fields

Abstract

Suppose $\mathbb{F}$ is a field of prime characteristic $p$ and $E$ is a finite subgroup of the additive group $(\mathbb{F},+)$. Then $E$ is an elementary abelian $p$-group. We consider two such subgroups, say $E$ and $E^{\prime }$, to be equivalent if there is an $\alpha \in {\mathbb{F}}^{\ast }:=\mathbb{F}\setminus \{0\}$ such that $E=\alpha E^{\prime }$. In this paper we show that rational functions can be used to distinguish equivalence classes of subgroups and, for subgroups of prime rank or rank less than twelve, we give explicit finite sets of separating invariants.

There has been a minor change of title from the previous version. We have expanded the introduction, simplified some of the proofs and corrected a number of minor errors. The document is now 25 pages