In 1878, Jordan showed that a finite subgroup of 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 was obtained when , achieved by the symmetric group Sn+1. In this paper, analogous bounds are obtained for the finite subgroups of the complex symplectic and orthogonal groups. In the case of the optimal bound is , achieved by the wreath product acting naturally on the direct sum of n 2-dimensional spaces; for the orthogonal groups , the generic linear group bound of is achieved as soon as .
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