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Unsolvable Problems in Groups With Solvable Word Problem

Published online by Cambridge University Press:  20 November 2018

James McCool*
Affiliation:
University of Toronto, Toronto, Ontario
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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].

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1970

References

1. Baumslag, G., Boone, W. W., and Neumann, B. H., Some unsolvable problems about elements and subgroups of groups, Math. Scand. 7 (1959), 191201.Google Scholar
2. Britton, J. L., Solution of the word problem for certain types of groups. I, Proc. Glasgow Math. Assoc. 3 (1956), 4554.Google Scholar
3. Clapham, C. R. J., Finitely presented groups with word problems of arbitrary degree of insolubility, Proc. London Math. Soc. (3) 14 (1964), 633676.Google Scholar
4. Clapham, C. R. J., An embedding theorem for finitely presented groups, Proc. London Math. Soc. (3) 17 (1967), 419430.Google Scholar
5. Fridman, A. A., On the relation between the word problem and the conjugacy problem infinitely defined groups, Trudy Moskov. Mat. Obsc. 9 (1960), 329365. (Russian)Google Scholar
6. Higman, G., Subgroups of finitely presented groups, Proc. Roy. Soc. Ser. A 262 (1961), 455475.Google Scholar
7. McCool, J., The order problem and the power problem for free product sixth-groups, Glasgow Math. J. 10 (1969), 19.Google Scholar
8. McCool, J., Embedding theorems for countable groups, Can. J. Math. 22 (1970), 827835.Google Scholar
9. Novikov, P. S., Unsolvability of the conjugacy problem in the theory of groups, Izv. Akad. Nauk SSSR Ser. Mat. 18 (1954), 485524. (Russian)Google Scholar
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