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Rigidity of \(p\)-adic \(p\)-torsion. (English) Zbl 0647.20007

This is a beautiful paper! It has its origin in the so-called Brauer-Zassenhaus conjecture asking whether, given a finite subgroup \(S\) of units in the integral group ring \({\mathbb{Z}}G\) of a finite group \(G\), there exists an invertible element \(u\) in the rational group algebra \({\mathbb{Q}}G\) that conjugates \(S\) into \(\pm G\). As is shown, for \(p\)-groups \(G\) the answer is positive in an even stronger sense, namely if one replaces \({\mathbb{Z}}\) by the \(p\)-adics, \({\mathbb{Z}}_ p\), then the \(u\) can already be taken in \({\mathbb{Z}}_ pG\). Very closely related to this is the work of K. Roggenkamp and L. Scott [Ann. Math., II. Ser. 126, 593-647 (1987; Zbl 0633.20003)]; in their paper, however, the groups \(S\) are always of the same size as \(G\). While they are using methods from, essentially, K-theory, in the present paper the result comes as a corollary of an important theorem on permutation modules which really is the heart of the matter. Namely, if \(G\) is a \(p\)-group and \(M\) a \({\mathbb{Z}}_ pG\)-lattice whose restriction to some normal subgroup \(N\) of \(G\) is free and which is such that \(M^ N\) appears as a permutation lattice for \(G/N\), then \(M\) itself is a permutation lattice. Actually, as a consequence one not only obtains the already mentioned result but also a generalization concerning matrices over \({\mathbb{Z}}_ pG\) which have also been studied in [Z. Marciniak, the reviewer, S. K. Sehgal and A. Weiss, J. Number Theory 25, 340-352 (1987; Zbl 0611.16007)].
Reviewer: J.Ritter

MSC:

20C05 Group rings of finite groups and their modules (group-theoretic aspects)
16S34 Group rings
16U60 Units, groups of units (associative rings and algebras)
20D15 Finite nilpotent groups, \(p\)-groups
20C11 \(p\)-adic representations of finite groups
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