Orbit sizes and the dihedral group of order eight
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Orbit sizes and the dihedral group of order eight Nathan A. Jones1 · Thomas Michael Keller2 Received: 7 November 2019 / Accepted: 29 February 2020 © Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In previous work of the second author with Yong Yang, it was shown that if G is a finite linear group which acts faithfully and completely reducibly on the finite vector space V, then |G∕G� | ≤ M , where M is the largest orbit size of G on V. Ii is also known that if |G∕G� | = M and G is nonabelian, then G is nilpotent and has at least two orbits of size M on V. In this paper, we study the latter situation in the extreme case that G has exactly two orbits of size M on V and that V is irreducible. It turns out that there is exactly one such action, namely when G is the dihedral group of order 8 acting on the vector space of order 9. Keywords Finite group · Orbit structure · Commutator subgroup · Dihedral group Mathematics Subject Classification 20D10
1 Introduction The study of the orbit structure in finite linear group actions has been a central topic of interest in recent years, and this paper is another contribution to this field of study. It con‑ tinues the study of the abelian quotient of a finite linear group and its relationship to the largest orbit size in the action. In [6], the second author and Y. Yang established the follow‑ ing result.
Theorem 1.1 [6] Let G be a finite solvable group and V a finite faithful completely reducible G-module, possibly of mixed characteristic. Let M be the largest orbit size in the action of G on V. Then
* Thomas Michael Keller [email protected] Nathan A. Jones [email protected] 1
Department of Mathematics, The University of Oklahoma, 601 Elm Avenue, Room 423, Norman, OK 73019‑3103, USA
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Department of Mathematics, Texas State University, 601 University Drive, San Marcos, TX 78666, USA
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N. A. Jones, T. M. Keller
|G∕G� | ≤ M. More precisely, we have one of the following: (a) |G∕G� | < M ; (b) |G∕G� | = M and G is abelian; or (c) |G∕G� | = M , G is nilpotent, and G has at least two different orbits of size M on V. The main conclusion of the above result, namely that |G∕G� | ≤ M , has since been gen‑ eralized to arbitrary finite groups, see [5]. The full conclusion of Theorem 1.1 is true for arbitrary finite groups by very recent work of Qian and Yang [10]. In this paper, we will focus on part (c) of the above theorem in the situation where V is irreducible as G-module and consider the “smallest” case when there are exactly two orbits of size M and when V is assumed to be irreducible. We are able to completely determine the group action in this case, and it turns out to be a well-known “small group action.” Our main result is as follows. Theorem 1.2 Let G be a finite nonabelian group and V a finite faithful irreducible G-module. Suppose that M = |G∕G� | is the largest orbit size of G on V and that there are exactly two orbits of size M on V. Then G is dihedral of order 8,
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