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The normalizer of a metabelian group in its integral group ring. (English) Zbl 1017.16023
The paper contributes results on the problem when the normalizer $$N_U(G)$$ of $$G$$ in the group $$U$$ of units of the integral group ring $$\mathbb{Z} G$$ of a finite group $$G$$ satisfies $$N_U(G)=G\cdot Z(U)$$, where $$Z(U)$$ is the centre of $$U$$. By definition, an automorphism $$\rho$$ of $$G$$ is called a Coleman automorphism (CA) if $$\rho$$ preserves the conjugacy classes of $$G$$ and is conjugation with some $$g_P\in G$$ on the Sylow subgroups $$P$$ of $$G$$, and if $$\rho^2$$ is inner.
Theorems: 1. Let $$G$$ be metabelian, $$B\rightarrowtail G\twoheadrightarrow A$$. If $$P=B_2\rtimes D$$ and $$C_{A_2}(B_2)=C_{A_2}(b_2)$$ for some $$b_2\in B_2$$, where $$P$$ is a Sylow 2-subgroup of $$G$$ and $$B_2$$, $$A_2$$ are such of $$B$$ and $$A$$, respectively, then every CA of $$G$$ is inner; in particular, $$N_U(G)=G\cdot Z(U)$$. 2. If the Sylow 2-subgroup of $$A$$ is cyclic, then every CA is again inner. (Above, $$C$$ denotes centralizers.) It follows that $$N_U(G)=G\cdot Z(U)$$ for split metabelian groups $$G$$ with a dihedral Sylow 2-subgroup.
Related papers are [M. Hertweck, Habilitation thesis, Stuttgart (2003); Y. Li, M. M. Parmenter and S. K. Sehgal, Commun. Algebra 27, No. 9, 4217-4223 (1999; Zbl 0943.16012); Z. S. Marciniak and K. W. Roggenkamp, NATO Sci. Ser. II, Math. Phys. Chem. 28, 159-188 (2001; Zbl 0989.20002); M. Mazur, J. Algebra 212, No. 1, 175-189 (1999; Zbl 0921.16018)].

##### MSC:
 16U60 Units, groups of units (associative rings and algebras) 20C05 Group rings of finite groups and their modules (group-theoretic aspects) 16S34 Group rings 20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, $$\pi$$-length, ranks 20D20 Sylow subgroups, Sylow properties, $$\pi$$-groups, $$\pi$$-structure
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