zbMATH — the first resource for mathematics

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

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
PDF BibTeX Cite
Full Text: DOI
[1] Brandt, J, A lower bound for the number of irreducible characters in a block, J. algebra, 74, 509-515, (1982) · Zbl 0478.20009
[2] Dornhoff, L, Group representation theory, (1972), Dekker New York, Part B · Zbl 0236.20004
[3] Külshammer, B, On the structure of block ideals in group algebras of finite groups, Comm. algebra, 8, 1867-1872, (1980) · Zbl 0446.20004
[4] Külshammer, B, Bemerkungen über die gruppenalgebra als symmetrische algebra, J. algebra, 72, 1-7, (1981) · Zbl 0472.16007
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.