On filtered multiplicative bases of group algebras
by
Victor Bovdi
University of Debrecen, 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:

1.

if b₁, b₂ in B then either b₁b₂=0 or b₁b₂ in B;

2.

B \cap \rad(A) is a K-basis for \rad(A), where \rad(A) denotes the Jacobson radical of A.

In [1] it was 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 [1]: When does a filtered multiplicative K-basis exist in the group algebra KG?

Note that by Higman's theorem the group algebra KG over a field of characteristic p has only finitely many isomorphism classes of indecomposable KG-modules if and only if all the Sylow p-subgroups of G are cyclic.

Let G be a finite abelian p-group. Then G=<a₁>×<a₂>× ... ×<a_s> is the direct product of cyclic groups <a_i> of order q_i, the set

B={(a1-1)n1(a2-1)n2 ... (as-1)ns     |     0 <= ni < qi}

is a filtered multiplicative K-basis of the group algebra KG over the field K of characteristic p.

P.Landrock and G.O.Michler [5] 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 [6] first showed examples of noncommutative group algebras KG, which have a filtered multiplicative K-bases.

In [2] and [3] we gave an explicit list of all metacyclic p-groups G and explicit list of all p-groups H with the cyclic subgroup of index p², such that the group algebras over the field K, KG and KH, respectively, have a filtered multiplicative K-basis. We also showed that in the case when G is either a powerful p-group or a 2-generated p-group ( p is odd) with the central cyclic commutator subgroup the group algebra KG over the field K of characteristic p does not have such K-basis.

In [4] we study this question in the case when G is either a two generated 2-group with cyclic commutator subgroup or |G| <= p⁵.

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

[2] Bovdi, V., On filtered multiplicative basis of the group algebras, Arch. Math. (Basel), 74, 2000 p.81-88

[3] Bovdi, V., On filtered multiplicative basis of the group algebras II, Preprint, 2000, p.1-16

[4] Bovdi, V. , Balogh, Zs., On filtered multiplicative basis of the group algebras III, Preprint, 2000, p.1-17

[5] Landrock, P., Michler, G.O. , Block structure of the smallest Janko group , Math. Ann., 232(3), 1978, p.205-238

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

Date received: June 28, 2000