Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-22T04:23:58.065Z Has data issue: false hasContentIssue false
  • English
  • Français

A family of groups with nice word problems

Published online by Cambridge University Press:  09 April 2009

Verena Huber Dyson
Verena Huber Dyson
Affiliation:
University of Illinois, Chicago Circle, Chicago, Illinois, 60680, U.S.A.
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This paper is an outgrowth of my old battle with the open sentence problem for the theory of finite groups. The unsolvability of the word problem for groups (cf. [1] and [4]) entails the undecidability of the open sentence problem for the elementary theory of groups and thus strengthens the original undecidability result for this theory (cf. [7]). The fact that the elementary theory of finite groups is also undecidable (cf. [2] and [6]) therefore justifies my interest in the open sentence problem for that theory. This paper contains a construction of groups that might lead to a negative solution.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 17 , Issue 4 , June 1974 , pp. 414 - 425
Copyright
Copyright © Australian Mathematical Society 1974

References

[1]Boone, W., ‘Certain simple unsolvable problems of group theory’, Nedere. Akad. Wetensch. Math. 16 (1954), 213–237, 492–497 and 17 (1955), 252–256, 571–577.Google Scholar
[2]Cobham, A.. ‘Undecidability in group theory’, Amer. Math. Soc. Notices 9 (1961), 406.Google Scholar
[3]Hall, P. ‘Finiteness conditions for soluble groups’, Proc. Roy. Soc. (3) 4 (1954), 419–436.Google Scholar
[4]Higman, G., ‘Subgroups of finitely presented groups’, Proc. Roy. Soc. A 262 (1961), 455–475.Google Scholar
[5]Huber-Dyson, V., ‘The word problem and residually finite groups’, Amer. Math. Soc. Notices 11 (1964), 743.Google Scholar
[6]Mal'cev, A. I., ‘The undecidability of the elementary theory of finite groups’, Dok Ak Nauk (1961), 1009–1012.Google Scholar
[7]Tarski, A., Mostowski, A. and Robinson, R., Undecidable Theories, (North-Holland Publishing Co., Amsterdam, 1963.)Google Scholar