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Note on the abstract group (2,3,7;9)

Published online by Cambridge University Press:  24 October 2008

John Leech
John Leech
Affiliation:
University of Glasgow
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Extract

The abstract group

is finite for n = 4,6,7,8, and the relations are incompatible for n = 1,2,3,5. A criterion of Coxeter ((1)) suggests that (2,3,7; n) should be infinite for all n ≥ 9, but its applicability to these groups is unproved, and it is not known whether there are any further examples of finite groups (2,3,7; n). However, (2,3,7; 9) has been proved infinite by Sims ((3)), and it follows at once that (2,3,7; n) is infinite whenever n is a multiple of 9 as it then has an infinite factor group.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 62 , Issue 1 , January 1966 , pp. 7 - 10
Copyright
Copyright © Cambridge Philosophical Society 1966

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References

REFERENCES

(1)Coxeter, H. S. M.The abstract group G 3,7,16. Proc. Edinburgh Math. Soc. (2), 13 (1962), 47–61.CrossRefGoogle Scholar
(2)Leech, J.Generators for certain normal subgroups of (2, 3, 7). Proc. Cambridge Philos. Soc. 61 (1965), 321–332.Google Scholar
(3)Sims, C. C.On the group (2, 3, 7; 9). Notices American Math. Soc. 11 (1964), 687–688.Google Scholar