Degenerations and orbits in finite abelian groups

TL;DR

This paper introduces a new way to classify elements in finite abelian groups using a degeneration concept, leading to a lattice structure that describes orbits and their enumeration, independent of prime p.

Contribution

It develops a novel degeneration framework to parametrize orbits in finite abelian groups via a distributive lattice, providing explicit enumeration and polynomial formulas for orbit sizes.

Findings

Orbit classification via a finite distributive lattice

Orbit sizes are polynomial in p

Orbit structure is independent of p for fixed type

Abstract

A notion of degeneration of elements in groups is introduced. It is used to parametrize the orbits in a finite abelian group under its full automorphism group by a finite distributive lattice. A pictorial description of this lattice leads to an intuitive self-contained exposition of some of the basic facts concerning these orbits, including their enumeration. Given a partition $\lambda$, the lattice parametrizing orbits in a finite abelian p-group of type $\lambda$ is found to be independent of p. The order of the orbit corresponding to each parameter, which turns out to be a polynomial in p, is calculated. The description of orbits is extended to subquotients by certain characteristic subgroups. Each such characteristic subquotient is shown to have a unique maximal orbit.