Jennings' theorem for p-split groups

Abstract

Jennings [Trans. Amer. Math. Soc. 50 (1941) 175–185] used the descending central series of dimension subgroups to produce a basis of the radical layers of the group ring of a p-group over a field k of characteristic p and, with it, a Hilbert polynomial for the dimensions of these layers. Later, Quillen [J. Algebra 10 (1968) 411–418] employed Lie ring methods of Lazard [Ann. Sci. Ecole Norm. Sup. (3) 71 (1954) 101–190] to refine the proof of Jennings' theorem. Analogously, Alperin [Quart. J. Math. Oxford Ser. (2) 39 (154) (1988) 129–133] extended Jennings' methods to permutation modules for p-groups, while Shalev [Proc. Amer. Math. Soc. 120 (2) (1994) 333–337] later recast this in terms of Quillen's formulation. All of this naturally extends to p-split groups, finite groups with normal p-Sylow. Consider the filtration of the group ring kG by the powers of its Jacobson radical. When G is a p-group, Quillen showed that the associated graded ring is isomorphic to the graded p-restricted universal enveloping algebra UJ ∗ ⊗ k ( G ) for the graded p-restricted Lie algebra generated from the filtration of G by its descending central series of dimension subgroups. For a p-split group G with p-Sylow P and a given maximal subgroup A of order prime to p, the associated graded ring is isomorphic to the skew group ring UJ ∗ ⊗ k ( P ) ⋊ A (Theorem 4), where A is concentrated in degree zero and the action of A on UJ ∗ ⊗ k ( P ) is induced from its conjugation action on P. Let H be a subgroup of G and S a semisimple kH-module. Consider the graded module associated to the filtration by the powers of the Jacobson radical of the module formed by inducing S up to PH. This graded module will be isomorphic to the Hopf algebra tensor product of S (regarded as a semisimple k P H -module) with the quotient of UJ ∗ ⊗ k ( P ) by a left ideal generated by elements corresponding to q − 1 , where the q are elements of the p-Sylow of H; inducing this graded module up to G will preserve the grading (Theorem 5). Hilbert polynomials for the Brauer character of the radical layers allow explicit computation of the isomorphism classes of the radical layers of the module produced by inducing S to G.