On a filtered multiplicative basis of group algebras
by
Victor Bovdi
Institute of Mathematics and Informatics, Kossuth University, Hungary

Let A be a finite-dimensional algebra over a field K and let B be a K-basis of A. Suppose that B has the following properties: if b₁, b₂ in B then either b₁b₂=0 or b₁b₂ in B; and B \cap rad(A) is a K-basis for rad(A), where rad(A) denotes the Jacobson radical of A. Then B is called a filtered multiplicative K-basis of A.

The filtered multiplicative K-basis arises in the theory of representations of algebras and was introduced first by H. Kupisch. In R. Bautista, P. Gabriel, A. Roiter and L. Salmeron proved that if there are only finitely many isomorphism classes of indecomposable A-modules over an algebraically closed field K, then A has a filtered multiplicative K-basis.

We study the following question from : When does exist a filtered multiplicative K-basis in the group algebra KG?

P. Lendrock and G.O. Michler proved that the group algebra of the smallest Janko group over a field of characteristic 2 does not have a filtered multiplicative K-basis. L. Paris gave examples of group algebras KG, which have no filtered multiplicative K-bases. He also showed that if K is a field of characteristic 2 and either a) G is a quaternion group of order 8 and also K contains a primitive cube root of the unity or b) G is a dihedral 2-group, then KG has a filtered multiplicative K-basis. We show that for the class of all metacyclic groups the groups mentioned in the items a) and b) are exactly those, for whose group algebras L.Paris in presented examples of multiplicative K-bases.

We also study this question for the group algebra KG over a field of characteristic p of the two-generated p-group G with a cyclic commutator subgroup.

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Bautista, R., Gabriel, P., Roiter, A., and Salmeron, L., Representation-finite algebras and multiplicative bases Invent.-Math. 81 (2) 1985 217-285

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Lendrock, P., Michler, G.O., Block structure of the Smallest Janko group Math. Ann. 232 (3) 1978 205-238

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Paris, L., Some examples of group algebras without filtred multiplicative basis L'Enseignement Math. 33 1987 307-314

Date received: November 9, 1998