Abstract
Let G be a finite group, and let V be a completely reducible faithful Gmodule. It has been known for a long time that if G is abelian, then G has a regular orbit on V. In this paper we generalize this result as follows. Assuming G to be solvable, we show that G has an orbit of size at least |G/G′| on V. This also strengthens a result of Aschbacher and Guralnick in that situation. Additionally, we prove a similar generalization of the well-known result that if G is nilpotent, then G has an orbit of size at least \(\sqrt {\left| G \right|} \) on V.
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Keller, T.M., Yang, Y. Abelian quotients and orbit sizes of solvable linear groups. Isr. J. Math. 211, 23–44 (2016). https://doi.org/10.1007/s11856-015-1259-4
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DOI: https://doi.org/10.1007/s11856-015-1259-4