Commuting probabilities of finite groups

Sean Eberhard*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

25 Citations (Scopus)


The commuting probability of a finite group is defined to be the probability that two randomly chosen group elements commute. Let P > (0,1] be the set of commuting probabilities of all finite groups. We prove that every point of P is nearly an Egyptian fraction of bounded complexity. As a corollary, we deduce two conjectures of Keith Joseph from 1977: all limit points of P are rational, and P is well ordered by >. We also prove analogous theorems for bilinear maps of abelian groups.

Original languageEnglish
Pages (from-to)796-808
Number of pages13
JournalBulletin of the London Mathematical Society
Issue number5
Early online date25 Jul 2015
Publication statusPublished - Oct 2015
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2015 London Mathematical Society.

ASJC Scopus subject areas

  • General Mathematics


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