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Generators for Alternating and Symmetric Groups

Published online by Cambridge University Press:  09 April 2009

I. M. S. Dey  and
James Wiegold
I. M. S. Dey
Affiliation:
The Open UniversityWalton, Bletchley, Bucks, U.K.
James Wiegold
Affiliation:
University College Cardiff
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Let Γ denote the modular group, that is, the free product of a group of order 2 and a group of order 3. Morris Newman investigates in [2] the factor-groups of Γ and calls them Γ-groups for short; thus a group is a Γ-group if and only if it has a generating set consisting of an element of order dividing 2 and an element of order dividing 3. Newman's interest centres on finite simple Γ-groups. He proves that the linear fractional groups LF(2,p) for primes p are Γ -groups, and poses the problem of deciding which of the alternating groups enjoy this property.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 12 , Issue 1 , February 1971 , pp. 63 - 68
Copyright
Copyright © Australian Mathematical Society 1971

References

[1]Miller, G. A., ‘On the groups generated by two operators’, Bull. Amer. Math. Soc. 7 (1901), 424–426.CrossRefGoogle Scholar
[2]Newman, Morris, ‘Maximal normal subgroups of the modular group’, Proc. Amer. Math. Soc. 19 (1968), 1138–1144.CrossRefGoogle Scholar
[3]Wielandt, Helmut, Finite permutation groups (Academic Press, New York, 1964).Google Scholar