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Bounds for finite semiprimitive permutation groups: order, base size, and minimal degree

Part of: Permutation groups

Published online by Cambridge University Press:  05 November 2020

Luke Morgan ,
Cheryl E. Praeger  and
Kyle Rosa
Luke Morgan
Affiliation:
University of Primorska, UP IAM, Muzejski trg 2, Koper6000, Slovenia University of Primorska, UP FAMNIT, Glagoljaška 8, Koper6000, Slovenia (luke.morgan@famnit.upr.si)
Cheryl E. Praeger
Affiliation:
Department of Mathematics and Statistics, Centre for the Mathematics of Symmetry and Computation, The University of Western Australia, 35 Stirling Highway, Crawley, WA6009, Australia (cheryl.praeger@uwa.edu.au; kylerosa@gmail.com)
Kyle Rosa
Affiliation:
Department of Mathematics and Statistics, Centre for the Mathematics of Symmetry and Computation, The University of Western Australia, 35 Stirling Highway, Crawley, WA6009, Australia (cheryl.praeger@uwa.edu.au; kylerosa@gmail.com)
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Abstract

In this paper, we study finite semiprimitive permutation groups, that is, groups in which each normal subgroup is transitive or semiregular. These groups have recently been investigated in terms of their abstract structure, in a similar way to the O'Nan–Scott Theorem for primitive groups. Our goal here is to explore aspects of such groups which may be useful in place of precise structural information. We give bounds on the order, base size, minimal degree, fixed point ratio, and chief length of an arbitrary finite semiprimitive group in terms of its degree. To establish these bounds, we study the structure of a finite semiprimitive group that induces the alternating or symmetric group on the set of orbits of an intransitive minimal normal subgroup.

Keywords

Permutation groupsbase sizeminimal degree

MSC classification

Primary: 20B05: General theory for finite groups
Secondary: 20B15: Primitive groups
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 63 , Issue 4 , November 2020 , pp. 1071 - 1091
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Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

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