Abstract
Let V be a non-trivial finite-dimensional vector space over a finite field F of characteristic p and let G be an irreducible subgroup of \(GL(V)\) having nilpotence class at most two. We prove that if \(|G|> |V|/2\), then G is cyclic, or \(|V|=3^2\) or \(5^2\). This is a refinement of Glauberman’s result for the tight bound of linear groups of nilpotence class two.
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References
Glauberman, G.: On Burnside’s other \({p^{a}}{q^{b}}\)-theorem. Pac. J. Math. 56(2), 469–476 (1975)
Gorenstein, D.: Finite Groups. Chelsea, New York (1980)
Huppert. B.: Endliche Gruppen I. Die Grundlehren der Mathematischen Wissenschaften, vol. 134. Springer, Berlin, New-York (1967)
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This research is sponsored by the Shanghai Sailing Program (20YF1413400) and the Young Scientists Fund of NSFC (12001359).
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Meng, H. Large linear groups of nilpotence class two. Arch. Math. 116, 363–367 (2021). https://doi.org/10.1007/s00013-020-01552-2
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DOI: https://doi.org/10.1007/s00013-020-01552-2