M. A. Mikheenko, “<nobr>$p$</nobr>-Nonsingular systems of equations over solvable groups”, Mat. Sb., 2024, Volume 215, Number 6,Pages <nobr>61

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$p$-Nonsingular systems of equations over solvable groups

M. A. Mikheenko^ab

^a Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia
^b Moscow Center of Fundamental and Applied Mathematics, Moscow, Russia

Abstract: Any group that has a subnormal series all factors in which are abelian and all factors except the last one are $p'$-torsion free, can be embedded into a group with a subnormal series of the same length, with the same properties and such that any $p$-nonsingular system of equations over this group is solvable in this group itself. Using this we prove that the minimal order of a metabelian group over which there exists a unimodular equation that is unsolvable in metabelian groups is $42$.
Bibliography: 14 titles.

Keywords: equations over groups, group rings, solvable groups.

MSC: Primary 20F70; Secondary 16S34, 16S50

Received: 06.10.2023 and 30.03.2024

DOI: 10.4213/sm10009