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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

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