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Sylow subgroups of transitive permutation groups II

Published online by Cambridge University Press:  09 April 2009

Cheryl E. Praeger
Cheryl E. Praeger
Affiliation:
Department of Mathematics, University of Western Australia, Nedlands, Australia, 6009.
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Abstract

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Let G be a transitive permutation group on a finite set of n points, and let P be a Sylow p-subgroup of G for some prime p dividing |G|. We are concerned with finding a bound for the number f of points of the set fixed by P. Of all the orbits of P of length greater than one, suppose that the ones of minimal length have length q, and suppose that there are k orbits of P of length q. We show that f ≦ kp − ip(n), where ip(n) is the integer satisfying 1 ≦ ip(n) ≦ p and n + ip(n) ≡ 0(mod p). This is a generalisation of a bound found by Marcel Herzog and the author, and this new bound is better whenever P has an orbit of length greater than the minimal length q.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 23 , Issue 3 , May 1977 , pp. 329 - 332
Copyright
Copyright © Australian Mathematical Society 1977

References

Herzog, Marcel and Praeger, Cheryl E. (1976), ‘On the fixed points of Sylow subgroups of transitive permutation groups’, J. Austral. Math. Soc. 21 (Series A), 428–437.CrossRefGoogle Scholar
Praeger, Cheryl E. (1973), ‘Sylow subgroups of transitive permutation groups’, Math. Z. 134, 179–180.CrossRefGoogle Scholar
Wielandt, Helmut (1964), Finite Permutation Groups (translated by Bercov, R.. Academic Press, New York, London, 1964).Google Scholar