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Class-preserving automorphisms and the normalizer property for Blackburn groups. (English) Zbl 1168.16017
Let \(G\) be a group and let \(\mathcal U\) be the group of units of the integral group ring \(\mathbb{Z} G\). The group \(G\) is said to have the ‘normalizer property’ if \(N_{\mathcal U}(G)=Z(\mathcal U)G\), where \(Z(\mathcal U)\) denotes the center of \(\mathcal U\) and \(N_{\mathcal U}(G)\) is the normalizer of \(G\) in \(\mathcal U\). If \(G\) has finite non-normal subgroups, then, following Blackburn, the authors denote by \(R(G)\) the intersection of all of them and say that \(R(G)\) is defined. The group \(G\) is said to be a ‘Blackburn group’ if \(R(G)\) is defined and non-trivial. The authors denote with \(\operatorname{Aut}_c(G)\) the group of class-preserving automorphisms of \(G\), and set \(\text{Out}_c(G)=\operatorname{Aut}_c(G)/\text{Inn}(G)\).
The paper is organized as follows: Section 1 is an introduction. In Section 2 the authors study class-preserving automorphisms.
The main results are: Proposition 2.7. Let \(G\) be a finite group having an Abelian normal subgroup \(A\) with cyclic quotient \(G/A\). Then class-preserving automorphisms of \(G\) are inner automorphisms.
Proposition 2.9. A class-preserving automorphism of a finite Blackburn group is an inner automorphism.
The main result of Section 3 is the following: Theorem 3.3. Suppose that \(G\) has a finite non-normal subgroup, and that \(R(G)\) is non-trivial. Then \(G\) has the normalizer property, \(N_{\mathcal U}(G)=Z(\mathcal U)G\).

MSC:
16U60 Units, groups of units (associative rings and algebras)
20E36 Automorphisms of infinite groups
16S34 Group rings
20C05 Group rings of finite groups and their modules (group-theoretic aspects)
20C07 Group rings of infinite groups and their modules (group-theoretic aspects)
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