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Symmetric local algebras and small blocks of finite groups. (English) Zbl 0567.20007
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.
Reviewer: W.Hamernik

##### MSC:
 20C20 Modular representations and characters 20C05 Group rings of finite groups and their modules (group-theoretic aspects) 16S34 Group rings 20D60 Arithmetic and combinatorial problems involving abstract finite groups
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