Modular analogues of Jordan's theorem for finite linear groups
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Michael J. Collins
Abstract
In 1878, Jordan [C. Jordan, Mémoire sur les equations différentielle linéaire à intégrale algébrique, J. reine angew. Math. 84 (1878), 89–215.] showed that a finite subgroup of GL(n, ℂ) contains an abelian normal subgroup whose index is bounded by a function of n alone. Previously, the author has given precise bounds [M. J. Collins, On Jordan's theorem for complex linear groups, J. Group Th. 10 (2007), 411–423.]. Here, we consider analogues for finite linear groups over algebraically closed fields of positive characteristic ℓ. A larger normal subgroup must be taken, to eliminate unipotent subgroups and groups of Lie type and characteristic ℓ, and we show that generically the bound is similar to that in characteristic 0—being (n + 1)!, or (n + 2)! when ℓ divides n + 2—given by the faithful representations of minimal degree of the symmetric groups. A complete answer for the optimal bounds is given for all degrees n and every characteristic ℓ.
© Walter de Gruyter Berlin · New York 2008
Artikel in diesem Heft
- Topology of negatively curved real affine algebraic surfaces
- The spinorial τ-invariant and 0-dimensional surgery
- Well-posedness, blow-up phenomena, and global solutions for the b-equation
- Siegel disks and periodic rays of entire functions
- Isomorphisms between Leavitt algebras and their matrix rings
- The number of smallest parts in the partitions of n
- Modular analogues of Jordan's theorem for finite linear groups
- Prime specialization in higher genus I
- Filling inequalities do not depend on topology
- Erratum to "Modular curves and Ramanujan's continued fraction"
Artikel in diesem Heft
- Topology of negatively curved real affine algebraic surfaces
- The spinorial τ-invariant and 0-dimensional surgery
- Well-posedness, blow-up phenomena, and global solutions for the b-equation
- Siegel disks and periodic rays of entire functions
- Isomorphisms between Leavitt algebras and their matrix rings
- The number of smallest parts in the partitions of n
- Modular analogues of Jordan's theorem for finite linear groups
- Prime specialization in higher genus I
- Filling inequalities do not depend on topology
- Erratum to "Modular curves and Ramanujan's continued fraction"
