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ON MINIMAL DEGREES OF QUOTIENT GROUPS

Part of: Algebraic combinatorics Permutation groups Representation theory of groups

Published online by Cambridge University Press:  14 October 2024

IBRAHIM ALOTAIBI
IBRAHIM ALOTAIBI*
Affiliation:
School of Mathematics, University of Sydney, New South Wales 2006, Australia
*
e-mail: ialo9634@uni.sydney.edu.au, ialotaibi@tvtc.gov.sa
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Abstract

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Keywords

permutation groupsquotient groupsminimal degreesexceptional groupsalmost exceptional groupssemidirect productswreath product

MSC classification

Primary: 20C99: None of the above, but in this section
Secondary: 05E10: Combinatorial aspects of representation theory 20B05: General theory for finite groups 20C30: Representations of finite symmetric groups

Information

Type
PhD Abstract
Information
Bulletin of the Australian Mathematical Society , Volume 111 , Issue 1 , February 2025 , pp. 177 - 179
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

For a finite group G, the minimal faithful permutation representation degree, denoted by $\mu (G)$ , is defined as the smallest $n \in \{0, 1, 2, \ldots \}$ such that G embeds in $\mbox {Sym}(n)$ . The task of determining $\mu (G)$ for an arbitrary G is a complex undertaking, and can be linked to addressing a difficult minimisation problem concerning the lattice of subgroups of G (see [Reference Easdown and Praeger11, Reference Johnson17]). It is interesting to note that the relationship between the minimal degrees of quotient groups and their parent groups is quite uncertain. Despite the fact that the quotient group may be simpler than the parent group, its lattice of subgroups may be more restrictive so that, when solving the minimisation problem, the minimal degree of the quotient group can actually be greater than the minimal degree of the parent group [Reference Neumann23]. In such cases, the parent group is called exceptional. Though exceptional groups are not particularly rare, this terminology, introduced in the 1980s in [Reference Easdown and Praeger11], has persisted and this class of groups has been explored by a number of researchers (see [Reference Chamberlain7, Reference Easdown and Praeger11, Reference Lemieux21, Reference Lemieux22]).

In the dissertation, we study the delicate relationship between the minimal degrees of finite groups and their respective quotient groups. We address some gaps in the current literature, rectify some existing flaws, and introduce new terminologies and directions for future research. The thesis is a blend of mathematical argument and concrete examples, supported by the use of computer algebra software (in particular, [Reference Bosma, Cannon and Playoust4, Reference Dokchitser9, 28]).

In Chapter 1, we provide an overview of the historical context of the concepts presented within the thesis, and highlight the contributions of researchers to this evolving field (in particular, [Reference Cooperstein8–Reference Holt and Walton16, Reference Karpilovsky18–Reference Kovacs and Praeger20, Reference Saunders24–Reference Saunders27, Reference Wright29]), establishing the framework for the study that follows. In Chapter 2, we provide some essential background information for the subsequent chapters. We also revisit the work of Lemieux [Reference Lemieux21, Reference Lemieux22] and provide a more accurate classification of the minimal degrees of groups of order $p^4$ , where p is an odd prime. We conclude this chapter by establishing an explicit isomorphism between some groups from Burnside’s 1911 classification of groups of order $p^4$ [Reference Burnside6] and some groups appearing in a 2017 paper by Britnel et al. [Reference Britnell, Saunders and Skyner5], in the context of the classification of exceptional groups of order $p^5$ . In Chapter 3, we reproduce a class of exceptional groups of order $p^5$ , described in the aforementioned 2017 paper, but providing more direct and simplified proofs. This is followed by Chapter 4, where we present a comprehensive classification of all exceptional groups of order $243 = 3^5$ , providing detailed explanations and proofs. This study serves as an adjustment to and amplification of a table that was previously published in [Reference Britnell, Saunders and Skyner5]. In Chapter 5, we investigate minimal degrees of groups associated with certain wreath products. We also construct sequences of groups that have the property where some proper quotients are isomorphic to subgroups that have the same minimal degree, thus possessing what is known as the almost exceptional property which arises in the context of the abelian quotients conjecture [Reference Kovacs and Praeger19, Reference Kovacs and Praeger20]. In addition, we demonstrate the possibility of having an almost exceptional group with an arbitrarily long chain of normal subgroups such that all of their respective quotients have the same minimal degree. Furthermore, it is possible to have an unlimited number of pairwise incomparable subgroups with the same minimal degree. The results depend on a theory of semidirect products where the base group is a vector space of dimension k over a field with p elements, with p a prime and k a positive integer. The base group is extended by a cyclic group of order p, which is represented by a $k \times k$ matrix, adapting a technique from [Reference Easdown and Hendriksen10, Reference Hendriksen15]. A final application is made to construct sequences of groups with the property that the direct products have minimal degrees that grow linearly with the number n of factors, while their respective quotients, realised as central products, have minimal degrees that grow exponentially with n. This generalises a result of Neumann [Reference Neumann23].

The thesis is supported by three appendices. In Appendix A, we provide detailed information about minimal degrees of groups of order at most 63, extending a table in [Reference Easdown and Saunders12]. In Appendix B, we provide a comprehensive analysis of the minimal degrees of groups of order 243 and their corresponding quotient groups of order 81. In Appendix C, we present a table of wreath product groups of various orders, up to 500, including their minimal degrees.

Some of this research is available in [Reference Alotaibi and Easdown1–Reference Alotaibi and Easdown3].

Footnotes

Thesis submitted to the University of Sydney in December 2023; degree awarded on 17 June 2024; supervisor David Easdown.

References

Alotaibi, I. and Easdown, D., ‘Minimal degrees of groups associated with some wreath products’, Preprint, August 2023 (School of Mathematics and Statistics, University of Sydney).Google Scholar
Alotaibi, I. and Easdown, D., ‘Exceptional groups of order 243’, Preprint, November 2023 (School of Mathematics and Statistics, University of Sydney).Google Scholar
Alotaibi, I. and Easdown, D., ‘Minimal degrees of groups of order 64 and relationships with minimal degrees of their quotients’, Preprint, in preparation.Google Scholar
Bosma, W., Cannon, J. and Playoust, C., ‘The Magma algebra system. I: the user language’, J. Symbolic Comput. 24 (1997), 235–265.CrossRefGoogle Scholar
Britnell, J. R., Saunders, N. and Skyner, T., ‘On exceptional groups of order ${p}^5$ ’, J. Pure Appl. Algebra 221 (2017), 2647–2665.CrossRefGoogle Scholar
Burnside, W., Theory of Groups of Finite Order (Cambridge University Press, Cambridge, 1911).Google Scholar
Chamberlain, R., ‘Minimal exceptional $p$ -groups’, Bull. Aust. Math. Soc. 98 (2018), 434–438.CrossRefGoogle Scholar
Cooperstein, B., ‘Minimal degree for a permutation representation of a classical group’, Israel J. Math. 30 (1978), 215–235.CrossRefGoogle Scholar
Dokchitser, T., Group Names, available online at https://people.maths.bris.ac.uk/~matyd/GroupNames/.Google Scholar
Easdown, D. and Hendriksen, M., ‘Minimal permutation representations of semidirect products of groups’, J. Group Theory 19(6) (2016), 1017–1048.CrossRefGoogle Scholar
Easdown, D. and Praeger, C. E., ‘On minimal permutation representations of finite groups’, Bull. Aust. Math. Soc. 38 (1988), 207–220.CrossRefGoogle Scholar
Easdown, D. and Saunders, N., ‘The minimal faithful permutation degree for a direct product obeying an inequality condition’, Comm. Algebra 44(8) (2016), 3518–3537.CrossRefGoogle Scholar
Elias, B., Minimally Faithful Group Actions and $p$ -Groups, Senior Thesis (Princeton University, 2005).Google Scholar
Elias, B., Silbermann, L. and Takloo-Bighash, R., ‘Minimal permutation representations of nilpotent groups’, Exp. Math. 19(1) (2010), 121–128.CrossRefGoogle Scholar
Hendriksen, M., Minimal Permutation Representations of Classes of Semidirect Products of Groups, MSc Thesis (University of Sydney, 2015).Google Scholar
Holt, D. F. and Walton, J., ‘Representing the quotient groups of a finite permutation group’, J. Algebra 248(1) (2002), 307–333.CrossRefGoogle Scholar
Johnson, D. L., ‘Minimal permutation representations of finite groups’, Amer. J. Math. 93 (1971), 857–866.CrossRefGoogle Scholar
Karpilovsky, G. I., ‘The least degree of a faithful representation of abelian groups’, Vestnik Khar’kov Gos. Univ. 53 (1970), 107–115.Google Scholar
Kovacs, L. and Praeger, C. E., ‘Finite permutation groups with large abelian quotients’, Pacific J. Math. 136 (1989), 283–292.CrossRefGoogle Scholar
Kovacs, L. and Praeger, C. E., ‘On minimal faithful permutation representations of finite groups’, Bull. Aust. Math. Soc. 62 (2000), 311–317.CrossRefGoogle Scholar
Lemieux, S., Minimal Degree of Faithful Permutation Representations of Finite Groups, MSc Thesis (Carleton University, Ottawa, 1999).Google Scholar
Lemieux, S., ‘Finite exceptional $p$ -groups of small order’, Comm. Algebra 35 (2007), 1890–1894.CrossRefGoogle Scholar
Neumann, P. M., ‘Some algorithms for computing with finite permutation groups’, in: Proceedings of Groups–St Andrews, 1985, London Mathematical Society Lecture Notes Series, 121 (eds. E. F. Robertson and C. M. Campell) (Cambridge University Press, Cambridge, 1987), 59–92.CrossRefGoogle Scholar
Saunders, N., ‘Minimal faithful permutation degrees of finite groups’, Aust. Math. Soc. Gaz. 35(5) (2008), 332–338.Google Scholar
Saunders, N., ‘The minimal degree for a class of finite complex reflection groups’, J. Algebra 323(3) (2010), 561–573.CrossRefGoogle Scholar
Saunders, N., Minimal Faithful Permutation Representations of Finite Groups, PhD Thesis (University of Sydney, 2011).Google Scholar
Saunders, N., ‘Minimal faithful permutation degrees for irreducible Coxeter groups and binary polyhedral groups’, J. Group Theory 17(5) (2014), 805–832.CrossRefGoogle Scholar
The GAP Group, GAP—Groups, Algorithms and Programming, Version 4.12.2; 2022. https://www.gap-system.org.Google Scholar
Wright, D., ‘Degrees of minimal embeddings for some direct products’, Amer. J. Math. 97 (1975), 897–903.CrossRefGoogle Scholar