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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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##### References:
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