Minimal permutation representations of finite simple classical groups. Special linear, symplectic, and unitary groups
A faithful permutation representation of least degree for a group \(G\) is called a minimal permutation representation of \(G\). A classical group is a general linear, or symplectic, or unitary, or orthogonal group over a field; a simple classical group is one isomorphic to the (unique) nonabelian composition factor of some classical group. Minimal permutation representations of finite sporadic simple groups have been much studied: their degrees and subdegrees were calculated, stabilizers of one and two points were found. The goal of this paper is to add to this study by treating finite simple classical groups. The degrees of minimal permutation representations of finite classical groups were found by \textit{B. N. Cooperstein} [in Isr. J. Math. 30, 213-235 (1978; Zbl 0383.20027)]. Some of the degrees listed therein are, however, not minimal, with inaccuracies arising from the violation of some of the inequalities for small fields or dimensions. Fortunately, the errors were readily amenable to corrections, and a revised list of minimal degrees appeared [in \textit{P. Kleidman} and \textit{M. Liebeck}, The subgroup structure of the finite classical groups (1990; Zbl 0697.20004), p. 175]. Unfortunately, neither is this list free of errors, namely, for groups \(U_{2m}(2)\), where \(m\) is not divisible by 3, and for \(P\Omega_{2m}^+(3)=O^+_{2m}(3)\), \(m\geq 4\), the minimal degrees are indicated incorrectly.
scientific article; zbMATH DE number 738949