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

Elena K. Brusyanskaya, Anton A. Klyachko and Andrey V. Vasil’ev

Vol. 302 (2019), No. 2, 437–452

DOI: 10.2140/pjm.2019.302.437

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.

Keywords

equations in groups, Frobenius's theorem, Solomon's theorem, Iwasaki's theorem

Mathematical Subject Classification 2010

Primary: 20D60, 20F70

Milestones

Received: 11 July 2018

Revised: 12 January 2019

Accepted: 20 January 2019

Published: 27 November 2019