Thep-Intersection Subgroups in Quasi-Simple and Almost Simple Finite Groups

AbstractLetpbe a fixed prime andGa finite group. A proper subgroupX<Gis called ap-intersection subgroupifX∩Xgis ap-group for eachg∈G\X, butXis not ap-group. In this paper we classify thep-intersection subgroups in the quasi-simple and almost simple finite groups

On Generators of Modular Invariant Rings of Finite Groups

Let G be a finite group, let V be an F G-module of finite dimension d, and denote by fi(V; G) the minimal number m such that the invariant ring S(V ) G is generated by finitely many elements of degree at most m. A classical result of E. Noether says that fi(V; G) jGj provided that char F is coprime to jGj!. If char F divides jGj then no bounds for fi(V; G) are known except for very special choices of G. In this paper we present a constructive proof of the following: If H G with [G : H] 2 F and if the restriction V jH is a permutation module (e.g. if V is a projective F G - module and H 2 Syl p (G)) then fi(V; G) maxfjGj; d(jGj \Gamma 1)g regardless of char F . 1 Introduction For any (unitary) commutative ring R let RMod denote the category of left R - modules. Let G be a finite group, R a commutative ring and RG the group ring of G over R. With RGmod we denote the full subcategory of RGMod consisting of modules V such that VR is also a free R - module of finite rank. Con..