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

TL;DR

This paper studies finite subgroups of fields with prime characteristic, showing how rational functions can distinguish their equivalence classes and providing explicit invariants for certain ranks.

Contribution

It introduces a method using rational functions to classify elementary abelian p-groups in fields and supplies explicit invariants for ranks up to twelve.

Findings

Rational functions distinguish subgroup equivalence classes.

Explicit separating invariants are provided for ranks less than twelve.

The approach applies to elementary abelian p-groups in fields of prime characteristic.

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 $E'$, to be equivalent if there is an $\alpha\in\mathbb{F}^*:=\mathbb{F}\setminus\{0\}$ such that $E=\alpha E'$. 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.