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Commutative group algebras of summable \(p\)-groups. (English) Zbl 1122.20003

Summary: 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 \(FG\cong FH\) as \(F\)-algebras for some group \(H\) and if \(G_p\) is summable, then \(H_p\) is summablale. These achievements improve results due to P. Hill and W. Ullery [Commun. Algebra 25, No. 12, 4029-4038 (1997; Zbl 0901.16012)] and also their generalizations given by P. Danchev [Hokkaido Math. J. 29, No. 2, 255-262 (2000; Zbl 0967.20003); Commun. Algebra 28, No. 5, 2521-2531 (2000; Zbl 0958.20003); ibid. 29, No. 5, 1953-1958 (2001; Zbl 0993.20003); Kyungpook Math. J. 44, No. 1, 21-29 (2004; Zbl 1060.20006)].

MSC:

20C07 Group rings of infinite groups and their modules (group-theoretic aspects)
16U60 Units, groups of units (associative rings and algebras)
16S34 Group rings
20K10 Torsion groups, primary groups and generalized primary groups
20K21 Mixed groups
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