Linear group actions with at most three orbits of the largest size

Let G be a finite group and let V be a finite completely reducible faithful G -module, possibly of mixed characteristic. The second author and Yong Yang proved in previous work that | G : G ′ | ≤ M , where M is the largest orbit size of G on V . They also showed that if G is a nonabelian group and |...

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Bibliographic Details
Published in:Monatshefte für Mathematik 2024-02, Vol.203 (2), p.313-322
Main Authors: Çınarcı, Burcu, Keller, Thomas Michael
Format: Article
Language:English
Subjects:
Group theory
Mathematics
Mathematics and Statistics
Orbits
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Summary:Let G be a finite group and let V be a finite completely reducible faithful G -module, possibly of mixed characteristic. The second author and Yong Yang proved in previous work that | G : G ′ | ≤ M , where M is the largest orbit size of G on V . They also showed that if G is a nonabelian group and | G : G ′ | = M , then G is nilpotent having at least two orbits of maximal size M on V . In this paper, we deal with the latter case, where G is nonabelian and | G : G ′ | = M , and we classify all linear group actions in which G has at most three orbits of size M on V .
ISSN:0026-9255
1436-5081
DOI:10.1007/s00605-023-01850-1