Strange divisibility in groups and rings

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.

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

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