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**The direct factor problem for modular Abelian group algebras.**
*(English)*
Zbl 0676.16010

Representation theory, group rings, and coding theory, Pap. in Honor of S. D. Berman, Contemp. Math. 93, 303-308 (1989).

[For the entire collection see Zbl 0666.00005.]

Let G be an abelian group and let F be a field of prime characteristic p. Denote by FG the group algebra of G over F and by V(FG) the group of normalized units of FG.

In this paper the author proves the following two important results. Theorem B. (i) If the torsion subgroup of G is the direct sum of cyclic p-groups, then G is a direct factor of V(FG) with complement a direct sum of cyclic p-groups. (ii) If G and F are both countable and if the torsion subgroup of G is a p-group, then G is a direct factor of V(FG) with complement a simply presented p-group. Theorem C. Let G be a p-group of cardinal not exceeding \(\aleph_ 1\) and p-length not exceeding \(\omega_ 1\). Then G is a direct factor of V(FG). If F is either perfect or countable, then the complement is simply presented.

Let G be an abelian group and let F be a field of prime characteristic p. Denote by FG the group algebra of G over F and by V(FG) the group of normalized units of FG.

In this paper the author proves the following two important results. Theorem B. (i) If the torsion subgroup of G is the direct sum of cyclic p-groups, then G is a direct factor of V(FG) with complement a direct sum of cyclic p-groups. (ii) If G and F are both countable and if the torsion subgroup of G is a p-group, then G is a direct factor of V(FG) with complement a simply presented p-group. Theorem C. Let G be a p-group of cardinal not exceeding \(\aleph_ 1\) and p-length not exceeding \(\omega_ 1\). Then G is a direct factor of V(FG). If F is either perfect or countable, then the complement is simply presented.

Reviewer: G.Karpilovsky

### MSC:

16U60 | Units, groups of units (associative rings and algebras) |

16S34 | Group rings |

20C07 | Group rings of infinite groups and their modules (group-theoretic aspects) |

20K10 | Torsion groups, primary groups and generalized primary groups |