Hertweck, Martin; Jespers, Eric Class-preserving automorphisms and the normalizer property for Blackburn groups. (English) Zbl 1168.16017 J. Group Theory 12, No. 1, 157-169 (2009). 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, \(N_{\mathcal U}(G)=Z(\mathcal U)G\). Reviewer: Nako Nachev (Plovdiv) Cited in 1 ReviewCited in 13 Documents 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) Keywords:class-preserving automorphisms; normalizer property; Blackburn groups PDF BibTeX XML Cite \textit{M. Hertweck} and \textit{E. Jespers}, J. Group Theory 12, No. 1, 157--169 (2009; Zbl 1168.16017) Full Text: DOI arXiv References: [1] DOI: 10.1016/0021-8693(66)90018-4 · Zbl 0141.02401 · doi:10.1016/0021-8693(66)90018-4 [2] DOI: 10.1017/S1446788700014051 · Zbl 1102.20023 · doi:10.1017/S1446788700014051 [3] DOI: 10.1006/jabr.2001.8760 · Zbl 0993.20017 · doi:10.1006/jabr.2001.8760 [4] DOI: 10.1515/JGT.2007.040 · Zbl 1124.16023 · doi:10.1515/JGT.2007.040 [5] DOI: 10.1006/jabr.2001.8724 · Zbl 1063.16036 · doi:10.1006/jabr.2001.8724 [6] DOI: 10.1080/00927879908826692 · Zbl 0943.16012 · doi:10.1080/00927879908826692 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.