Vol. 302, No. 2, 2019

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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
Abstract

Our result contains as special cases the Frobenius theorem (1895) on the number of solutions to the equation xn = 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
Authors
Elena K. Brusyanskaya
Faculty of Mechanics and Mathematics
Moscow State University
Moscow
Russia
Anton A. Klyachko
Faculty of Mechanics and Mathematics
Moscow State University
Moscow
Russia
Andrey V. Vasil’ev
Sobolev Institute of Mathematics
Novosibirsk
Russia
Novosibirsk State University
Novosibirsk
Russia