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Zassenhaus conjecture (ZC1) on torsion units of integral group rings for some metabelian groups. (English) Zbl 1099.16012

The authors extend the validity of (ZC1) for groups \(G=A\ltimes X\), with \(A\) cyclic normal, \(X\) Abelian and \(\gcd(|A|,|X|)=1\) to groups \(G=AX\) where now a single common prime divisor \(p\) in \(|A|\) and \(|X|\) is permitted whenever the Sylow-\(p\) subgroup \(A_p\) of \(A\) and the Hall-\(p'\) subgronp \(X_{p'}\) of \(X\) commute elementwise. The methods of proof depend on [G. Cliff and A. Weiss, Trans. Am. Math. Soc. 352, No. 1, 457-475 (2000; Zbl 0941.20002)] and [Z. Marciniak, J. Ritter, S. K. Sehgal and A. Weiss, J. Number Theory 25, 340-352 (1987; Zbl 0611.16007)].

MSC:

16U60 Units, groups of units (associative rings and algebras)
16S34 Group rings
20C05 Group rings of finite groups and their modules (group-theoretic aspects)
11R27 Units and factorization
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