Minimal permutation representations of finite simple classical groups. Special linear, symplectic, and unitary groups

Abstract

R,,, 6 V', then R~, ee (VAF°). Indeed, ifR~; 6 VAF0 ° and e(R~;) = R~;NF = R~,, then A0 C R,g, A0 C R~;, A~ C_ R,g, and m (R~,)[A0] C m(R,~), but rn0 .... ,rnn E m (R~,) and, consequently, 1 = ~ rnial 6 m(R.,L), i<_n an impossibility. The proof is complete. REFERENCES 1. Yu. L. Ershov, "Boolean families of valuation rings," Algebra Logika, 31, No. 3, 274-294 (1992).2. P. Roquette, "Principal ideal theorem for holomorphy rings in fields," J. Reine Ang. Math., 262/263, 361-374 (1973). Translated by O. Bessonova MINIMAL PERMUTATION REPRESENTATIONS OF FINITE SIMPLE CLASSICAL GROUPS. SPECIAL LINEAR, SYMPLECTIC, AND UNITARY GROUPS V. D. MazurovUDC 519.44A 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) nonabehan composition factor of some classical group.Minimal permutation representations of finite sporadic simple groups have been much studied to date: their degrees and subdegrees are calculated, stabilizers of one and two points are found (see the summary table in [1]). 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 Cooperstein in [2]. 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 [3, p. 175]. Unfortunately, neither is this list free of errors, namely, for groups U2,,,(2), where m is not divisible by 3, and for Pf~+,,(3) = O+,n(3), m > 4, the minimal degrees are indicated incorrectly.

Primitive parabolic permutation representations for finite symplectic groups

“…We know from [10] that the stabilizer G W of the subspace W is a maximal parabolic subgroup of G and that each maximal parabolic subgroup is the stabilizer of some isotropic subspace. Consider a representation of G in terms of permutations on a set Ω of left cosets of the group G with respect to a subgroup G W , in which every element g of G is assigned a permutation mapping each coset xG W into gxG W .…”

The Non-Solvable Rank 3 Affine Planes

“…Looking at the minimal degree for a non-trivial permutation representation of a classical group we obtain that in most cases the representation must be trivial and from this a contradiction follows by the existence of too large planar p-groups. Since the Cooperstein's paper [6] on the minimal degree for a non-trivial permutation representation of a classical group contains some inaccuracies, the Cooperstein's results are intended to be corrected comparing with [26,34,35].…”

Primitive parabolic permutation representations for finite simple orthogonal groups in odd dimensions

“…By now these parameters have been obtained for minimal faithful permutation representations of all finite simple groups of Lie type (see [1][2][3][4][5][6]). An important class of permutation representations for finite groups of Lie type is formed by parabolic representations, that is, representations on cosets with respect to parabolic subgroups.…”