Symmetric local algebras and small blocks of finite groups (Q1059712)

A familiar problem in modular representation theory is to determine the numbers of ordinary and modular irreducible characters belonging to a p- block B of a finite group G with defect group D. The author tackles the opposite question of classifying (in a special case) the defect group D by the numbers of ordinary and modular irreducible characters in B. Theorem: Let B be a block of the finite group G having only one modular and \(\leq 4\) ordinary irreducible characters. Then the order of the defect group D of B is equal to the number of ordinary irreducible characters in B (hence D is cyclic or a Klein four group). The proof does not use Dade's and Brauer's classifications of the blocks with cyclic or Klein four group defect groups. Rather it is a consequence (using a result of Brauer) of the following property of symmetric local algebras. Theorem: Let A be a symmetric algebra having a one-dimensional radical factor algebra and a centre of dimension \(\leq 4\). Then A is commutative.