Degenerations and orbits in finite abelian groups

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.

14 pages, 5 figures

References in corpus (2)

• On character values and decomposition of the Weil representation associated to a finite abelian group
• Characteristic Subgroups of Finite Abelian Groups

Cited by in corpus (4)

• Combinatorics of finite abelian groups and Weil representations
• Degeneration and orbits of tuples and subgroups in an Abelian group
• On the Positivity Conjecture for Finite Abelian p-Groups
• Permutation Representations of the Orbits of the Automorphism Group of a Finite Module over Discrete Valuation Ring