## Basic subgroups in modular Abelian group algebras.(English)Zbl 1167.16026

Summary: Suppose $$F$$ is a perfect field of $$\text{char\,}F=p\neq 0$$ and $$G$$ is an arbitrary Abelian multiplicative group with a $$p$$-basic subgroup $$B$$ and $$p$$-component $$G_p$$. Let $$FG$$ be the group algebra with normed group of all units $$V(FG)$$ and its Sylow $$p$$-subgroup $$S(FG)$$, and let $$I_p(FG;B)$$ be the nilradical of the relative augmentation ideal $$I(FG;B)$$ of $$FG$$ with respect to $$B$$.
The main results that motivate this article are that $$1+I_p(FG;B)$$ is basic in $$S(FG)$$, and $$B(1+I_p(FG;B))$$ is $$p$$-basic in $$V(FG)$$ provided $$G$$ is $$p$$-mixed. These achievements extend in some way a result of N. Nachev when $$G$$ is $$p$$-primary. Thus the problem of obtaining a ($$p$$-)basic subgroup in $$FG$$ is completely resolved provided that the field $$F$$ is perfect.
Moreover, it is shown that $$G_p(1+I_p(FG;B))/G_p$$ is basic in $$S(FG)/G_p$$, and $$G(1+I_p(FG;B))/G$$ is basic in $$V(FG)/G$$ provided $$G$$ is $$p$$-mixed. As consequences, $$S(FG)$$ and $$S(FG)/G_p$$ are both starred or divisible groups.

### MSC:

 16U60 Units, groups of units (associative rings and algebras) 20C07 Group rings of infinite groups and their modules (group-theoretic aspects) 20K27 Subgroups of abelian groups 20K10 Torsion groups, primary groups and generalized primary groups 20K20 Torsion-free groups, infinite rank 20K21 Mixed groups 16S34 Group rings
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