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Seminormal and subnormal subgroup lattices for transitive permutation groups

Part of: Permutation groups

Published online by Cambridge University Press:  09 April 2009

Cheryl E. Praeger
Cheryl E. Praeger
Affiliation:
School of Mathematics and Statistics, The University of Western Australia, 35 Stirling Highway, Crawley WA 6009, Australia, e-mail: praeger@maths.uwa.edu.au
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Abstract

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Various lattices of subgroups of a finite transitive permutation group G can be used to define a set of ‘basic’ permutation groups associated with G that are analogues of composition factors for abstract finite groups. In particular G can be embedded in an iterated wreath product of a chain of its associated basic permutation groups. The basic permutation groups corresponding to the lattice L of all subgroups of G containing a given point stabiliser are a set of primitive permutation groups. We introduce two new subgroup lattices contained in L, called the seminormal subgroup lattice and the subnormal subgroup lattice. For these lattices the basic permutation groups are quasiprimitive and innately transitive groups, respectively.

Keywords

transitive permutation groupprimitivequasiprimitiveinnately transitivesubgroup latticesubnormal subgroup.

MSC classification

Secondary: 20B05: General theory for finite groups 20B10: Characterization theorems 20B25: Finite automorphism groups of algebraic, geometric, or combinatorial structures
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 80 , Issue 1 , February 2006 , pp. 45 - 64
Copyright
Copyright © Australian Mathematical Society 2006

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