Yet another Freiheitssatz: Mating finite groups with locally indicable ones

Любую группу, обладающую субнормальным рядом, в котором все факторы абелевы и все, за исключением последнего, не имеют $p'$-кручения, можно вложить в группу с субнормальным рядом такой же длины и с такими же свойствами такую, что любая $p$-невырожденная система уравнений над этой группой разрешима в самой этой группе. Это позволяет доказать, что минимальный порядок метабелевой группы, над которой есть унимодулярное уравнение, не разрешимое в метабелевых группах, равен $42$. Библиография: 14 названий.

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$p$-Nonsingular systems of equations over solvable groups

Mikheenko

2024

Sb. Math.

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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.

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Yet another Freiheitssatz: Mating finite groups with locally indicable ones

Abstract: The main result includes as special cases on the one hand, the Gerstenhaber–Rothaus theorem (1962) and its generalisation due to Nitsche and Thom (2022) and, on the other hand, the Brodskii–Howie–Short theorem (1980–1984) generalising Magnus’s Freiheitssatz (1930).

Cited by 3 publications

References 33 publications

Equations over solvable groups

Equations over solvable groups

О $p$-невырожденных системах уравнений над разрешимыми группами

$p$-Nonsingular systems of equations over solvable groups

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