The Cohen-Lenstra Heuristic: Methodology and Results

TL;DR

This paper demonstrates the equivalence of two probability measures in number theory and group theory, enabling the transfer of results between the Cohen-Lenstra heuristic and matrix conjugacy class studies.

Contribution

It establishes the equivalence of two probability measures, bridging number theory and group theory, and provides a comprehensive survey of methods and results in both fields.

Findings

Proves the equivalence of Cohen-Lenstra measure and matrix conjugacy measure.

Enables transfer of results between number theory and group theory.

Provides a survey of existing methods and findings in both areas.

Abstract

In number theory, great efforts have been undertaken to study the Cohen-Lenstra probability measure on the set of all finite abelian $p$-groups. On the other hand, group theorists have studied a probability measure on the set of all partitions induced by the probability that a randomly chosen $n\times n$-matrix over $\FF_p$ is contained in a conjucagy class associated with this partitions, for $n \to \infty$. This paper shows that both probability measures are identical. As a consequence, a multitide of results can be transferred from each theory to the other one. The paper contains a survey about the known methods to study the probability measure and about the results that have been obtained so far, from both communities.