A note on a theorem of May concerning commutative group algebras

Abstract

Let G be a coproduct of p-primary abelian groups with each factor of cardinality not exceeding , , and let F be a perfect field of characteristic p . If V(G) is the group of normalized units of the group algebra F(G), it is shown that G is a direct factor of V(G) and that the complementary factor is simply presented. This generalizes a theorem of W. May, who proved the result in the case when G itself has cardinality not exceeding N, and length not exceeding co I Throughout let F denote a perfect field of characteristic p t 0 and let G denote a p-primary abelian group. In a recent paper, W. May [M2] proved the following result, where V(G) denotes the group of normalized units in the group algebra F(G); hence V(G) = { cigi E F(G): Ec = 1}. Theorem 1 ([M2]). If IGI < t1 and the length of G does not exceed wo1, then G is a direct factor of V(G) and the complementary factor is simply presented. Recall that a group is simply presented if it can be presented by generators and relations that involve at most two generators. The main purpose of this note is to show that the condition on the length of G in the above theorem can be omitted. In other words, the theorem is proved for a much more general class of groups (inasmuch as there exists an abundance of abelian p-groups of cardinality not exceeding t1 with arbitrary prescribed length i < wd2 ). Actually, we extend the theorem to any abelian p-group G which is a coproduct of groups with the cardinality of each factor not exceeding tj; of course such a G can have arbitrarily large cardinality. We refer to [MI] and [M3] as examples of the connection between the direct factor theorem and the isomorphism problem: When does F(G) F(G') imply that G G' ? As is customary for group algebras, we use the multiplicative notation for G even though G is abelian. Consequently, for an ordinal a, G' is the translation of the more familiar p7G; since there is only one relevant prime, Received by the editors November 28, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 20C)7; Secondary 20K10. Supported in part by NSF grant DMS 8800862. ? 1990 American Mathematical Society 0002-9939/90 $1.00 + $.25 per page