On finite Alperin 2-groups with elementary abelian second commutants

By an Alperin group we mean a group in which the commutant of each 2-generated subgroup is cyclic. Alperin proved that if p is an odd prime then all finite p-groups with this property are metabelian. The today??s actual problem is the construction of examples of nonmetabelian finite Alperin 2-groups. Note that the author had given some examples of finite Alperin 2-groups with second commutants isomorphic to Z ₂ and Z ₄ and proved the existence of finite Alperin 2-groups with cyclic second commutants of however large order by appropriate examples. In this article the existence is proved of finite Alperin 2-groups with abelian second commutants of however large rank.     

The inductive Alperin–McKay condition for 2-blocks with cyclic defect groups

We verify the inductive Alperin–McKay condition introduced by the second author, for 2-blocks of the covering groups of finite simple non-abelian groups with cyclic defect groups.     

On the second commutants of finite Alperin groups

We refer to an Alperin group as a group in which the commutant of every 2-generated subgroup is cyclic. Alperin proved that if p is an odd prime then all finite p-groups with the property are metabelian. Nevertheless, finite Alperin 2-groups may fail to be metabelian. We prove that for each finite abelian group H there exists a finite Alperin group G for which G″ is isomorphic to H.     

Finite alperin 2-groups with cyclic second commutants

An Alperin group is a group in which every 2-generated subgroup has a cyclic commutant. Previously, we constructed examples of finite Alperin 2-groups with second commutant isomorphic to Z ₂ or Z ₄. Here, it is proved that for any natural n, there exists a finite Alperin 2-group whose second commutant is isomorphic to Z _2n .     

Alperin’s fusion theorem for localities

In these notes we give a version of the Alperin–Goldschmidt fusion theorem for localities.     

The Alperin and Dade Conjectures for the O'Nan and Rudivalis Simple Groups

We classify the radical subgroups and chains of the O'Nan and Rudvalis simple groups O'N and Ru, and then verify the Alperin weight conjecture and the Dade final conjecture for the two groups.     

Finite groups with odd Sylow automizers

Let p be an odd prime number. In this paper, we characterize the nonabelian composition factors of a finite group with odd p-Sylow automizers, and then prove that the McKay conjecture, the Alperin weight conjecture, and the Alperin–McKay conjecture hold for such a group.     

The Alperin and dade conjectures for the simple Conway’s third group

This paper is part of a program to study Alperin’s weight conjecture and Dade’s conjecture on counting ordinary characters in blocks for several finite groups. The classifications of radical subgroups and certain radical chains and their local structures of the simple Conway’s third group have been obtained by using the computer algebra system CAYLEY. The Alperin weight conjecture and the Dade final conjecture have been confirmed for the group.     

On a Theorem of Barker–Puig that Refines Alperin’s Weight Conjecture for Blocks in p-Solvable Finite Groups via Clifford Theory

We present a much shorter Clifford Theoretic Proof of an important Theorem of Barker and Puig on a refinement of Alperin’s Weight Conjecture for Blocks in p-solvable finite groups. Our proof employs a standard Clifford Theoretic approach. We also demonstrate a relationship between the Green correspondence and Sibley’s concept of a vertex pair. Consequently the main theorem can be stated in terms of Sibley’s vertex pairs.     

The Alperin and Dade Conjectures for the Conway Simple Group Co1

We classify the radical subgroups and chains of the Conway simple group Co₁ and then verify the Alperin weight conjecture and the Dade final conjecture for this group.     

Brauer correspondence and characters of height zero of monomial groups

The Brauer correspondence between the p-blocks of the monomial groups GSym(n) and the normalizers of their Sylow p-subgroups is determined. It is also shown that corresponding blocks contain the same number of characters of height zero, if the same assertion holds for G. This varifies a conjecture of J.L. Alperin in this special case.     

On finite alperin <Emphasis Type="Italic">p</Emphasis>-groups with homocyclic commutator subgroup

We study metabelian Alperin groups, i.e., metabelian groups in which every 2-generated subgroup has a cyclic commutator subgroup. It is known that, if the minimum number d(G) of generators of a finite Alperin p-group G is n ≥ 3, then d(G′) ≤ C _n ² for p≠ 3 and d(G′) ≤ C _n ² + C _n ³ for p = 3. The first section of the paper deals with finite Alperin p-groups G with p≠ 3 and d(G) = n ≥ 3 that have a homocyclic commutator subgroup of rank C _n ² . In addition, a corollary is deduced for infinite Alperin p-groups. In the second section, we prove that, if G is a finite Alperin 3-group with homocyclic commutator subgroup G- of rank C _n ² + C _n ³ , then G″ is an elementary abelian group.     

Dade’s ordinary conjecture implies the Alperin–McKay conjecture

Archiv der Mathematik - We show that Dade’s ordinary conjecture implies the Alperin–McKay conjecture. We remark that some of the methods can be used to identify a canonical height zero...     

Controlled blocks of the finite quasisimple groups for odd primes

We classify controlled blocks, introduced by Alperin and Broué in 1979 for all quasisimple groups G for odd primes. The results imply that every nilpotent block of G has abelian defect groups, which in turn is one of the main results proved in An and Eaton (2011) [6]. We also give an explicit characterization of non-controlled blocks of all quasisimple groups G for odd primes. This implies the block theoretic analogue of Glauberman?s ZJ-theorem for G proved by Kessar, Linckelmann and Robinson (2002) [18].     

Conjectures on the character degrees of the harada-norton simple group HN

We classify the radical subgroups and chains of the Harada-Norton simple group HN and verify the Alperin weight conjecture and the refined Dade conjecture due to Uno for the group. This implies the Isaacs-Navarro and Dade reductive conjectures for the group. This work was supported in part by the Marsden Fund of New Zealand via grant #9144/3368248.     

A refinement of Alperin’s Conjecture for blocks of the endomorphism algebra of the Sylow permutation module

We present a refinement of Alperin’s Conjecture involving the blocks of the endomorphism algebra of the permutation module formed by the cosets of a p-subgroup. We prove the conjecture in two special cases where every weight module has a simple socle.     

On π-Subpairs of π-Blocks

Alperin and Broug have given the p-subpairs in a finite group, and proved that there is a Sylow theorem for p-subpairs. For a π-separable group with π-Hall subgroup nilpotent, we prove that there is a π-Sylow theorem for π-subpairs. Note that our π-subpairs are different from what Robinson and Staszewski gave.     

Relatively stable equivalences of Morita type for blocks

Based on the fact that the relatively stable category of a p-block B is equivalent to the relatively stable category of its Brauer correspondent b as triangulated category, we introduce the notion of relatively stable equivalence of Morita type and show that there is a relatively stable equivalence of Morita type between B and b. Some invariants under stable equivalence of Morita type can be generalized to this relative case. In particular, we put forward the generalized Alperin–Auslander conjecture and prove it in special cases.     

超聚焦子群是16阶初等交换群的块北大核心CSCD

假设有限群上的块b被其亏群的正规化子控制,其超聚焦子群是16阶的初等交换群并且包含在亏群的中心里.本文计算了部分情况下块b的不可约模特征标和不可约常特征标个数.在这些情况下,Alperin重量猜想是对的.     

Varietal isologism and covering groups

In this paper it will be shown that any two \bf\cal V-covering groups of a given group are V\bf\cal V-isologic with respect to the variety V\bf\cal V, which is a vast generalization of a result in B. Huppert (1967) and R. L. Griess JR (1973). We also give a criterion of existence of V\bf\cal V-covering groups for a V\bf\cal V-perfect group, and show that every automorphism of a given V\bf\cal V-perfect group G can be extended to an automorphism of the V\bf\cal V-covering G^* say, of G, this generalizes a result of J. L. Alperin and D. Gorenstein (1966), in the abelian variety.