Удк 512.742 Some Isomorphism Results on Commutative Group Algebras

Throughout the present paper, suppose FG is the group algebra of an abelian group G written multiplicatively (and possibly mixed) over a field F of positive characteristic p. As usual, FG denotes the group algebra of G over F with unit group U(FG) and normalized unit group V (FG); note that the direct decomposition U(FG) = V (FG) × F ∗ holds where F ∗ is the multiplicative group of F and thus the study of U(FG) reduces to the study of V (FG). Moreover, let S(FG) = Vp(FG) and let Gp be the p-primary components of V (FG) and G, respectively. All other notions and notations are standard and follow essentially those from the reference list at the end of the article. Nevertheless, we will give below some supplementary terminology and concepts. Two central subjects in commutative group algebras theory have been played a major role. First of all, this is the isomorphism problem for commutative modular group algebras which is still unresolved and seems insurmountable in full generality at this stage. It states as follows: Isomorphism Conjecture. Suppose G is a p-mixed group. If H is any group and FG ∼= FH as F -algebras, then H ∼= G. This question was settled by many authors for various classes of abelian groups; for example the interested reader can see [1–9] together with the complete references in [10–12]. Here we shall confirm once again its truthfulness provided G is splitting whose maximal torsion subgroup Gt is a totally projective (reduced or not) p-group (compare also with the original source [1]). Note that when the torsion subgroup Gt is a torsion-complete p-group, the readers may see [3], [5] and [8]. All of this is subsumed by the second challenging topic. Specifically, we state the following yet left-open. Direct Factor Conjecture. Let G be a p-mixed group and let F be perfect. Then V (FG)/G is simply presented and, in particular, G is a direct factor V (FG) with simply presented complement.