The Cohen-Lenstra Heuristic: Methodology and Results

2009/12/25 by Johannes Lengler, Lengler, Johannes
Mathematics · #Advanced Operator Algebra Research #Finite Group Theory Research #Limits and Structures in Graph Theory #math.GR #math.NT #math.PR #msc:11R11 #msc:11R29 #msc:15A21 #msc:15A33 #msc:20K01 #msc:20K30 #msc:60B15 #msc:60B20

paper · pdf · doi:10.48550/arxiv.0912.4975

arxiv created 2009/12/25 · arxiv updated 2010/01/14

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× n-matrix over \FF_p is contained in a conjucagy class associated with this partitions, for n → ∞. 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.

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