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Coprime subdegrees of twisted wreath permutation groups

Published online by Cambridge University Press:  28 June 2019

Alexander Y. Chua
Affiliation:
Department of Mathematics and Statistics, Centre for the Mathematics of Symmetry and Computation, The University of Western Australia, 35 Stirling Highway, Crawley, WA 6009, Australia (21506815@student.uwa.edu.au; michael.giudici@uwa.edu.au; luke.morgan@famnit.upr.si)
Michael Giudici
Affiliation:
Department of Mathematics and Statistics, Centre for the Mathematics of Symmetry and Computation, The University of Western Australia, 35 Stirling Highway, Crawley, WA 6009, Australia (21506815@student.uwa.edu.au; michael.giudici@uwa.edu.au; luke.morgan@famnit.upr.si)
Luke Morgan
Affiliation:
Department of Mathematics and Statistics, Centre for the Mathematics of Symmetry and Computation, The University of Western Australia, 35 Stirling Highway, Crawley, WA 6009, Australia (21506815@student.uwa.edu.au; michael.giudici@uwa.edu.au; luke.morgan@famnit.upr.si)

Abstract

Dolfi, Guralnick, Praeger and Spiga asked whether there exist infinitely many primitive groups of twisted wreath type with non-trivial coprime subdegrees. Here, we settle this question in the affirmative. We construct infinite families of primitive twisted wreath permutation groups with non-trivial coprime subdegrees. In particular, we define a primitive twisted wreath group G(m, q) constructed from the non-abelian simple group PSL(2, q) and a primitive permutation group of diagonal type with socle PSL(2, q)m, and determine many subdegrees for this group. A consequence is that we determine all values of m and q for which G(m, q) has non-trivial coprime subdegrees. In the case where m = 2 and $q\notin \{7,11,29\}$, we obtain a full classification of all pairs of non-trivial coprime subdegrees.

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 2019 

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