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Determining Subgroups of a Given Finite Index in a Finitely Presented Group

Published online by Cambridge University Press:  20 November 2018

Anke Dietze  and
Mary Schaps
Anke Dietze
Affiliation:
IBM, Hamburg, West Germany
Mary Schaps
Affiliation:
Tel-Aviv University, Tel-Aviv, Israel
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The use of computers to investigate groups has mainly been restricted to finite groups. In this work, a method is given for finding all subgroups of finite index in a given group, which works equally well for finite and for infinite groups. The basic object of study is the finite set of cosets. §2 reviews briefly the representation of a subgroup by permutations of its cosets, introduces the concept of normal coset numbering, due independently to M. Schaps and C. Sims, and describes a version of the Todd-Coxeter algorithm. §3 contains a version due to A. Dietze of a process which was communicated to J. Neubuser by C. Sims, as well as a proof that the process solves the problem stated in the title. A second such process, developed independently by M. Schaps, is described in §4. §5 gives a method for classifying the subgroups by conjugacy, and §6, a suggestion for generalization of the methods to permutation and matrix groups.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 26 , Issue 4 , August 1974 , pp. 769 - 782
Copyright
Copyright © Canadian Mathematical Society 1974

References

1. Dietze, A., Drei Vorfahren zur Bestimmung sämtlicher Untergruppen endlich prdsentierbarer Gruppen zu vorgegebenem Index (Diplomarbeit, Kiel, 1970).Google Scholar
2. Felsch, H., Programmierung der Restklassenabzdhlung eine Gruppe nach Untergruppen, Numer. Math. 3 (1961), 250–256.Google Scholar
3. Mendelsohn, N. S., An algorithmic solution for a word problem in group theory, Can. J. Math. 16 (1964), 509–516. Correction: Can. J. Math. 17 (1965), 505.Google Scholar
4. Schaps, M., An algorithm to generate subgroups of finite index in a group given by defining relations (Manuscript, Kiel, 1968).Google Scholar
5. Sims, C. C., Computational methods in the study of groups, conference on Computational Problems in Abstract Algebra, Oxford, 1967; private communications, Oxford, 1967, and Kiel, 1969.Google Scholar
6. Todd, J. and Coxeter, H. S. M., A practical method for enumerating cosets of a finite abstract group, Proc. Edinburgh Math. Soc. (1936).Google Scholar