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arXiv:1411.0848 [math.GR]AbstractReferencesReviewsResources

Commuting probabilities of finite groups

Sean Eberhard

Published 2014-11-04Version 1

The commuting probability of a finite group is defined to be the probability that two randomly chosen group elements commute. Let P \subset (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.

Comments: 14 pages
Categories: math.GR, math.NT
Subjects: 20D60, 20F24, 15A63
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