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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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