Strange divisibility in groups and rings
Group Theory
2017-05-02 v5 Rings and Algebras
Abstract
We prove a general divisibility theorem that implies, e.g., that, in any group, the number of generating pairs (as well as triples, etc.) is a multiple of the order of the commutator subgroup. Another corollary says that, in any associative ring, the number of Pythagorean triples (as well as four-tuples, etc.) of invertible elements is a multiple of the order of the multiplicative group.
Cite
@article{arxiv.1506.08967,
title = {Strange divisibility in groups and rings},
author = {Anton A. Klyachko and Anna A. Mkrtchyan},
journal= {arXiv preprint arXiv:1506.08967},
year = {2017}
}
Comments
7 pages. A Russian version of this paper is at http://halgebra.math.msu.su/staff/klyachko/papers.htm . V.2: misprints corrected, some applications generalised. V.3: minor additions and corrections. V.4: Some corrections and generalisations (see Theorem on Monomorphisms and Subgroups). V.5: A reference is added