Commutative Group Algebras of Summable p-Groups

Abstract

Let G be an Abelian group whose p-component G _p is summable and let F be a field of characteristic p. Let S(FG) be the normalized unit p-group of the group algebra FG. The main results of the present article are that G _p is a direct factor of S(FG) with totally projective complement, provided G _p is of countable length and F is perfect; and, in particular, under these circumstances S(FG) is summable if and only if G _p is summable. Moreover, if FG≅ FH as F-algebras for some group H and if G _p is summable, then H _p is summablale. These achievements improve results due to Hill and Ullery (1997) and also their generalizations given by Danchev (2000a b 2001 2004).

Keywords:

• Direct factors
• Isomorphisms
• Summable groups
• Unit groups

Mathematics Subject Classification:

• Primary 20C07
• Secondary 16U60, 16S34, 20K10, 20K21

ACKNOWLEDGMENTS

The author would like to express his warm gratitude to the expert referee for the constructive criticism, helpful comments, and suggestions made as well as to the Editor, Professor David Easdown, for the precise editorial control.

Notes

Communicated by D. Easdown.