UDC 512.54, 004.021
DOI: 10.36871/2618-9976.2023.04.003

Authors

Irina V. Dobrynina,
Doctor of Physical and Mathematical Sciences, Associate Professor, Professor of Department of Mathematical Analysis, Moscow Technical University of Communications and Informatics, Moscow, Russia
Evgeny V. Manokhin,
Head of the Department of Mathematics and Computer Science, Tula Branch of the Financial university, Tula, Russia

Abstract

At the beginning of the twentieth century, the fundamental problems in group theory from the point of view of the existence of algorithms took shape. These include the problems of words, as well as the conjugacy of arbitrary words (formulated by M. Den). In addition, the problem of isomorphism for groups is posed (formulated by G. Titze). Russian algebraists P.S. Novikov and S.I. Adyan were able to obtain a proof of the unsolvability of these problems for all groups belonging to finitely defined groups. After the publication of these results, algorithmic problems, together with their various generalizations, are investigated already in fixed classes of groups. As one of the generalizations of the problem for conjugacy of arbitrary words, the question of conjugacy of arbitrary subgroups is posed, namely, the question of finding an algorithm that, by arbitrary subgroups given by a finite set of generating and defining words of some group, could determine whether they are conjugate in this group or not conjugate. As another generalization of the conjugacy of words, we can consider generalized conjugacy for words, which allows us to determine an algorithm capable of determining from arbitrary finite sets of words from a certain group whether they will be conjugated in it or not. If both of these generalizations are combined into one, then we get a generalization of the conjugacy problem of subgroups. Artin groups appeared a long time ago, they include wellknown braid groups that have been studied from an algebraic point of view since the twenties of the last century. The solution of the problems under consideration in Artin's groups caused great difficulties, which led to the allocation of various subclasses. The article proves the solvability of generalization of the conjugacy problem of subgroups in Artin groups on two generating ones.

Keywords

Algorithm, Algorithmic problem, Artin group, Tree structure, Conjugate occurrence, Cyclic subgroup