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Permutation Groups with Multiply‐Transitive Suborbits, II

Permutation Groups with Multiply‐Transitive Suborbits, II PERMUTATION GROUPS WITH MULTIPLY-TRANSITIVE SUBORBITS, II PETER J. CAMERON In a previous paper with the same title [1], I considered the following situation; G is a primitive, not doubly transitive permutation group on Q, in which the stabiliser G of a point a acts doubly transitively on an orbit F(a), where |F(a)| = v. Manning [3] showed that, if v > 2, then G has an orbit larger than F(a). Indeed, with A = F* o T (see [1] for notation), it is easy to see that A(a) is a G -orbit and |A(a)| = v(v-l)/k, with k < v-l if v > 2. In [1], I showed that k < \{v-\) if v > 5. Furthermore, if G is triply transitive on F(a), then k = 0(i>*); if G is a a quadruply transitive on F(a), then k < 2. I remarked there that the truth is probably stronger that these results suggest, since essentially only two situations are known in which k > 2; these are the Mathieu group M (or its automorphism group) with v = 16, k = 4, |Q| = 77, and the Higman-Sims group HS (or its automorphism group) with v = 22, http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

Permutation Groups with Multiply‐Transitive Suborbits, II

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References (10)

Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/6.2.136
Publisher site
See Article on Publisher Site

Abstract

PERMUTATION GROUPS WITH MULTIPLY-TRANSITIVE SUBORBITS, II PETER J. CAMERON In a previous paper with the same title [1], I considered the following situation; G is a primitive, not doubly transitive permutation group on Q, in which the stabiliser G of a point a acts doubly transitively on an orbit F(a), where |F(a)| = v. Manning [3] showed that, if v > 2, then G has an orbit larger than F(a). Indeed, with A = F* o T (see [1] for notation), it is easy to see that A(a) is a G -orbit and |A(a)| = v(v-l)/k, with k < v-l if v > 2. In [1], I showed that k < \{v-\) if v > 5. Furthermore, if G is triply transitive on F(a), then k = 0(i>*); if G is a a quadruply transitive on F(a), then k < 2. I remarked there that the truth is probably stronger that these results suggest, since essentially only two situations are known in which k > 2; these are the Mathieu group M (or its automorphism group) with v = 16, k = 4, |Q| = 77, and the Higman-Sims group HS (or its automorphism group) with v = 22,

Journal

Bulletin of the London Mathematical Society – Wiley

Published: Jul 1, 1974

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