# zbMATH — the first resource for mathematics

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)
Full Text: