Unsolvable Problems in Groups With Solvable Word Problem

James McCool

doi:10.4153/CJM-1970-094-6

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Let G be a finitely presented group with solvable word problem. It is of some interest to ask which other decision problems must necessarily be solvable for such a group. Thus it is easy to see that there exist effective procedures to determine whether or not such a group is trivial, or nilpotent of a given class. On the other hand, the conjugacy problem need not be solvable for such a group, for Fridman [5] has shown that the word problem is solvable for the group with unsolvable conjugacy problem given by Novikov [9].

References

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9. Novikov, P. S., Unsolvability of the conjugacy problem in the theory of groups, Izv. Akad. Nauk SSSR Ser. Mat. 18 (1954), 485–524. (Russian)Google Scholar

Crossref Citations

This article has been cited by the following publications. This list is generated based on data provided by Crossref.

Noskov, G. A. Remeslennikov, V. N. and Roman'kov, V. A. 1982. Infinite groups. Journal of Soviet Mathematics, Vol. 18, Issue. 5, p. 669.

Remeslennikov, V. N. and Roman'kov, V. A. 1985. Model-theoretic and algorithmic questions in group theory. Journal of Soviet Mathematics, Vol. 31, Issue. 3, p. 2887.

Bokut', L. A. and Kukin, G. P. 1989. Unsolvable algorithmic problems for semigroups, groups, and rings. Journal of Soviet Mathematics, Vol. 45, Issue. 1, p. 871.

Anokhin, M. I. 1997. On the word problem and the conjugacy problem for groups of the formF/V(R). Mathematical Notes, Vol. 61, Issue. 1, p. 3.

Анохин, Михаил Игоревич and Anokhin, Mikhail Igorevich 1997. О проблемах равенства и сопряженности для групп вида $F/V(R)$. Математические заметки, Vol. 61, Issue. 1, p. 3.

OLSHANSKII, ALEXANDER YU. and SAPIR, MARK V. 2005. SUBGROUPS OF FINITELY PRESENTED GROUPS WITH SOLVABLE CONJUGACY PROBLEM. International Journal of Algebra and Computation, Vol. 15, Issue. 05n06, p. 1075.

Gillibert, Pierre 2018. An automaton group with undecidable order and Engel problems. Journal of Algebra, Vol. 497, Issue. , p. 363.

Bogopolski, Oleg and Ivanov, Aleksander 2022. Decidability problem for exponential equations in finitely presented groups. Canadian Mathematical Bulletin, p. 1.

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Unsolvable Problems in Groups With Solvable Word Problem

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