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On conjugacy of group bases of the integral group ring. (Russian) Zbl 0596.16007
Let G be a finite group and let U(\({\mathbb{Z}}G)\) be the unit group of the integral group ring \({\mathbb{Z}}G\). Denote by m the number of nonconjugate normalized group bases of \({\mathbb{Z}}G\). It is known that m is always finite [G. Karpilovsky, J. Aust. Math. Soc., Ser. A 29, 385-392 (1980; Zbl 0432.16007)]. If G is Abelian or a Hamiltonian 2-group, then \(m=1\) since in this case \({\mathbb{Z}}G\) has only trivial units of finite order. If G is dihedral of order 8, then \(m=2\) by virtue of a result of C. Polcino Milies [Bol. Soc. Bras. Mat. 4, 85-92 (1973; Zbl 0379.20004)].
The aim of this paper is to provide a class of groups for which \(m\geq 2\). The following is an example of such groups. Let \(G=<A,b>\) where A is an Abelian subgroup of G of exponent 4, \(b^ 2\in A\), \(b^{-1}ab=a^{-1}\) for all \(a\in A\) and G is not Hamiltonian (Theorem 2). Other examples are too technical to quote here.
Reviewer: G.Karpilovsky

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