On finite subgroups of the classical groups

Abstract

Abstract In 1878, Jordan showed that a finite subgroup of GL ( n , ℂ ) ${\operatorname{GL}(n,\mathbb {C})}$ must possess an abelian normal subgroup whose index is bounded by a function of n alone. In previous papers, the author obtained optimal bounds; in particular, a generic bound ( n + 1 ) ! ${(n+1)!}$ was obtained when n ≥ 71 ${n\ge 71}$ , achieved by the symmetric group S n+1. In this paper, analogous bounds are obtained for the finite subgroups of the complex symplectic and orthogonal groups. In the case of Sp ( 2 n , ℂ ) ${\operatorname{Sp}(2n,\mathbb {C})}$ the optimal bound is ( 60 ) n · n ! ${(60)^{n}\cdot n!}$ , achieved by the wreath product SL 2 ( 5 ) wr S n ${\operatorname{SL}_{2}(5)\operatorname{wr}S_{n}}$ acting naturally on the direct sum of n 2-dimensional spaces; for the orthogonal groups O ( n , ℂ ) ${\mathrm {O}(n,\mathbb {C})}$ , the generic linear group bound of ( n + 1 ) ! ${(n+1)!}$ is achieved as soon as n ≥ 25 ${n\ge 25}$ .