Strange divisibility in groups and rings

TL;DR

This paper proves a general divisibility theorem linking the number of generating sets in groups and Pythagorean triples in rings to subgroup orders, revealing underlying algebraic divisibility properties.

Contribution

It introduces a unifying divisibility theorem that connects generating sets in groups and Pythagorean triples in rings, providing new insights into algebraic structures.

Findings

Number of generating pairs in any group is divisible by the order of its commutator subgroup

Number of Pythagorean triples of invertible elements in rings is divisible by the order of the ring's multiplicative group

Establishes a broad divisibility principle applicable to various algebraic configurations

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.