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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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