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Theorems of Rietz and Wielandt on Transitive Permutation Groups

Theorems of Rietz and Wielandt on Transitive Permutation Groups THEOREMS OF RIETZ AND WIELANDT ON TRANSITIVE PERMUTATION GROUPS PETER M. NEUMANN 1. The context The theorems of Rietz (1904) and Wielandt (1962) that are to be extended in this note deal with a permutation group G acting transitively on a finite set Q. They give information about the behaviour of a stabiliser G of a point a of Q. RIETZ'S THEOREM [1, Theorem 10]. Suppose that G is primitive on ft and is finite of odd order. If G has just two orbits T and A in ft — {a}, then the action ofG on F is a a faithful. Recall that there is a natural pairing of G -in variant subsets of ft. It was originally discovered by Burnside in 1901, and, as was pointed out by Manning in 1927, may be described as a process of 'reflection' (see [3, §16]): if F is such a set we define K:={geG\a.€rg} and then define the reflection F* to be <xK, that is, {<xg\geK}. Another description, often more convenient, depends on the simple fact that there is a one-one correspondence between (/-invariant binary relations on ft (identified with subsets of ft x ft) and C? -invariant subsets of ft, http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

Theorems of Rietz and Wielandt on Transitive Permutation Groups

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/24.4.335
Publisher site
See Article on Publisher Site

Abstract

THEOREMS OF RIETZ AND WIELANDT ON TRANSITIVE PERMUTATION GROUPS PETER M. NEUMANN 1. The context The theorems of Rietz (1904) and Wielandt (1962) that are to be extended in this note deal with a permutation group G acting transitively on a finite set Q. They give information about the behaviour of a stabiliser G of a point a of Q. RIETZ'S THEOREM [1, Theorem 10]. Suppose that G is primitive on ft and is finite of odd order. If G has just two orbits T and A in ft — {a}, then the action ofG on F is a a faithful. Recall that there is a natural pairing of G -in variant subsets of ft. It was originally discovered by Burnside in 1901, and, as was pointed out by Manning in 1927, may be described as a process of 'reflection' (see [3, §16]): if F is such a set we define K:={geG\a.€rg} and then define the reflection F* to be <xK, that is, {<xg\geK}. Another description, often more convenient, depends on the simple fact that there is a one-one correspondence between (/-invariant binary relations on ft (identified with subsets of ft x ft) and C? -invariant subsets of ft,

Journal

Bulletin of the London Mathematical SocietyWiley

Published: Jul 1, 1992

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