Hyperfocal subalgebras of source algebras of blocks over small-ground fields

Abstract

1.1. The local theory of blocks of finite groups was proposed originally by J. Alperin and M. Broue in [1], and developed by L. Puig [12], where the source algebra of a block is introduced as the smallest algebra which carries the local information of the block. One of the classical applications of the theory is the research on nilpotent blocks (see [2,9]). Recently, understanding the fusions of local pointed groups, L. Puig in [7] and [8] introduces the hyperfocal subalgebra in the source algebra of a block, and proves its existence and uniqueness up to conjugation. The local information of nilpotent blocks are the simplest case, and the structure theorem of their source algebras in [9] is the simplest case of the Puig’s work on hyperfocal subalgebras. Noting that Puig obtains his results in large enough coefficient fields, in this paper we make a research on the hyperfocal subalgebras of source algebras of blocks over small ground-fields.