A new canonical induction formula for p-permutation modules

1.1. Let G be a connected reductive algebraic group defined over a finite (prime) field Fp of p elements. For a given rational G-module M one can consider the restrictions of M to the restricted Lie algebra Lie(G) and to the finite group G(Fp) of Fp-rational points. Even though there is no direct functorial connection between the categories of restricted Lie(G)-modules and G(Fp)-modules, the ambient algebraic group G can serve as a bridge linking these two important module categories. Early results of Curtis in the 1960s provided a one-to-one correspondence between the simple restricted modules (over k = Fp) of Lie(G) and those of G(Fp). This correspondence is given by simply restricting certain classes of simple G-modules. More importantly, these results, along with work of Steinberg, allow one to easily transfer the questions pertaining to the computation of irreducible characters among these three categories. For over thirty years, this approach has been one of the main ideas used to relate the representation and cohomology theory of algebraic groups, Lie algebras, and finite groups of Lie type (in the defining characteristic) (see, e.g., [CPSvan], [Jan2], [Hum3], and [Jan1]). Given the previous results in this area, a natural course of investigation should be to find relationships between canonical modules, like projective modules, for Lie(G) and G(Fp). More precisely,

If Λ \Lambda is an artin algebra there is a partition of ind ⁡ Λ \operatorname {ind} \Lambda , the category of indecomposable finitely generated Λ \Lambda -modules, ind ⁡ Λ = ∪ i ⩾ 0 P _ _ i \operatorname {ind} \Lambda = { \cup _{i \geqslant 0}}{\underline {\underline {P}}_i} , called the preprojective partition. We show that P _ _ i \underline {\underline {P}}_i can be easily constructed for hereditary artin algebras, if P _ _ i − 1 \underline {\underline {P}}_{i - 1} is known: A A is in P _ _ i \underline {\underline {P}}_i if and only if A A is not in P _ _ i − 1 \underline {\underline {P}}_{i - 1} and there is an irreducible map B → A B \to A , where B B is in P _ _ i − 1 \underline {\underline {P}}_{i - 1} .

Variants of some of the Brauer-Fowler theorems

Ehresmann semigroups whose categories are EI and their representation theory

The Loewy and socle structure of the projective indecomposable modules for the group algebras of the Mathieu group M 12 and its automorphism group M 12.2 over an algebraically closed field of characteristic 3 are determined without using computers.

In this paper we study projective modules over subrings of k [ X , Y ] k[X,Y] . Conditions are given for projective modules to decompose into free ⊕ \oplus rank 1 modules. Our main result is that if k is an algebraically closed field and A a subring of B = k [ X , Y ] B = k[X,Y] with A ⊂ B A \subset B integral and sing ( A ) {\text {sing}}(A) finite, then all f.g. projective A-modules have the form free ⊕ \oplus rank 1. We also give several examples of subrings of k [ X , Y ] k[X,Y] which have indecomposable projective modules of rank 2.

A new description of the characters of the protective indecomposable modules for SL(2,pm) in characteristic p is given.From this, one can get the Car tan invariants of these groups by computing the products of certain matrices.This method explains why previous complete descriptions of these invariants are so complicated.We also answer a problem of Upadhyaya concerning the total number of composition factors in one particular projective indecomposable module.

For a finite group $H$‎, ‎let $cs(H)$ denote the set of non-trivial conjugacy class sizes of $H$ and $OC(H)$ be the set of the order components of $H$‎. ‎In this paper‎, ‎we show that if $S$ is a finite simple group with the disconnected prime graph and $G$ is a finite group such that $cs(S)=cs(G)$‎, ‎then $|S|=|G/Z(G)|$ and $OC(S)=OC(G/Z(G))$‎. ‎In particular‎, ‎we show that for some finite simple group $S$‎, ‎$G cong S times Z(G)$‎.

A well-known theorem of Fong states that over large enough fields of any characteristic, the principal indecomposable modules of a soluble finite group are induced from subgroups of order prime to the characteristic.It is shown that this property in fact characterises soluble finite groups.

Blocks with a cyclic defect group

Blocks whose defect groups are Klein four groups

For an action of a finite cyclic group F on an n -dimensional noncommutative torus A_\theta , we give sufficient conditions when the fundamental projective modules over A_\theta , which determine the range of the canonical trace on A_\theta , extend to projective modules over the crossed product C^* -algebra A_\theta \rtimes F . Our results allow us to understand the range of the canonical trace on A_\theta \rtimes F , and determine it completely for several examples including the crossed products of 2-dimensional noncommutative tori with finite cyclic groups and the flip action of \mathbb Z_2 on any n -dimensional noncommutative torus. As an application, for the flip action of \mathbb Z_2 on a simple n -dimensional torus A_\theta , we determine the Morita equivalence class of A_\theta \rtimes \mathbb Z_2, in terms of the Morita equivalence class of $A_\theta.

The solubility of a finite group with less than [Formula: see text] supersoluble subgroups is confirmed in this paper. Moreover, we prove that a finite insoluble group has exactly [Formula: see text] supersoluble subgroups if and only if it is isomorphic to [Formula: see text]. Furthermore, it is shown that a finite group with less than [Formula: see text] conjugacy classes of supersoluble subgroups is soluble, and the only finite insoluble group with [Formula: see text] conjugacy classes of supersoluble subgroups is isomorphic to [Formula: see text].

The Loewy structure of projective indecomposable modules for some modular group algebras

We show that, for any positive integer k, there are only finitely many finite groups, up to isomorphism, with exactly k conjugacy classes of elements of prime power order. This generalizes a result of E. Landau from 1903. The proof of our generalization makes use of the classification of finite simple groups.