What do Frobenius's, Solomon's, and Iwasaki's theorems on divisibility in groups have in common?

TL;DR

This paper unifies and generalizes classical theorems on divisibility and solutions in groups, revealing common underlying principles and deriving new corollaries for groups and rings.

Contribution

It presents a unifying framework that encompasses Frobenius, Solomon, and Iwasaki theorems, offering new insights into divisibility properties in algebraic structures.

Findings

Unified theorem covering Frobenius, Solomon, and Iwasaki results

Derived new corollaries for groups and rings

Provided a generalized approach to solutions of equations in groups

Abstract

Our result contains as special cases the Frobenius theorem (1895) on the number of solutions to the equation $x^n=1$ in a group, the Solomon theorem (1969) on the number of solutions in a group to a system of equations having fewer equations than unknowns, and the Iwasaki theorem (1985) on roots of subgroups. There are other curious corollaries on groups and rings.