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Some definitions of Klein's simple group of order 168 and other groups

Published online by Cambridge University Press:  18 May 2009

John Leech
John Leech
Affiliation:
The University, Glasgow
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In a previous note [3], Mennicke and I showed that the relations

(8, 7|2, 3): A8=B7=(AB)2=(AB)2=(A-1B)2=(A-1B)3=E

define a group of order 10752. As we remarked, the results of §§ 2, 3 of that note are not restricted in their application to this group; they apply to the group

[3, 7]+: B7=(AB)2=(A-1B)3=E

and to any factor group of this group which in turn has Klein's simple group of order 168, defined by

(4,7|2, 3): A4=B7=(AB)2=(A-1B)3=E,

as a factor group. In this note I use these results to establish alternative “weaker” definitions for Klein's group and for two groups discussed by Sinkov [4], namely (8, 7|2, 3) defined above and a factor group of this group of order 1344. These latter groups are eloquently discussed by Coxeter [1].

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 5 , Issue 4 , July 1962 , pp. 166 - 175
Copyright
Copyright © Glasgow Mathematical Journal Trust 1962

References

1.Coxeter, H. S. M., The abstract group G3, 7, 15, Proc. Edinburgh Math. Soc. (2) 12 (1962).Google Scholar
2.Coxeter, H. S. M. and Moser, W. O. J., Generators and relations for abstract groups (Ergebnisse dcr Math. N.F. 14, Springer, 1957), Chapter 2.CrossRefGoogle Scholar
3.Leech, J. and Mennicke, J., Note on a conjecture of Coxeter, Proc. Glasgow Math. Assoc. 5, (1961), 25–29.CrossRefGoogle Scholar
4.Sinkov, A., On the group-defining relations (2, 3, 7; p), Ann. of Math. (2) 38 (1937), 577–584.CrossRefGoogle Scholar
5.Todd, J. A. and Coxeter, H. S. M., A practical method for enumerating cosets of a finite abstract group, Proc. Edinburgh Math. Soc. (2) 5 (1936), 26–34.CrossRefGoogle Scholar