Search

Suppose that all nontrivial subsections of a p -block B are conjugate (where p is a prime). By using the classification of the finite simple groups, we prove that the defect groups of B are either extraspecial of order p 3 with p ∈ { 3 , 5 } or elementary abelian. [ABSTRACT FROM AUTHOR]

Abstract: We introduce a stronger and more tractable form of Oliverʼs p-group conjecture, and derive a reformulation in terms of the modular representation theory of a quotient group. The Sylow p-subgroups of the symmetric group and of the general linear group satisfy both the strong conjecture and its reformulation. [Copyright &y& Elsevier]

Abstract: In this paper we investigate general properties of Cartan invariants of a finite group G in characteristic 2. One of our results shows that the Cartan matrix of G in characteristic 2 contains an odd diagonal entry if and only if G contains a real element of 2-defect zero. We also apply these results to 2-blocks of symmetric groups and to blocks with normal or abelian defect groups. The second part of the paper deals with annihilators of certain ideals in centers of group algebras and blocks. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate certain ideals in the center of a symmetric algebra A over an algebraically closed field of characteristic . These ideals include the Higman ideal and the Reynolds ideal. They are closely related to the p-power map on A. We generalize some results concerning these ideals from group algebras to symmetric algebras, and we obtain some new results as well. In case , these ideals detect odd diagonal entries in the Cartan matrix of A. In a sequel to this paper, we will apply our results to group algebras. [Copyright &y& Elsevier]