# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] Coleman, D.B., On the modular group ring of a p-group, Proc. amer. math. soc., 15, 511-514, (1964) · Zbl 0132.27501 [2] Hertweck, M., A counterexample to the isomorphism problem for integral group rings of finite groups, Ann. of math., 154, 115-138, (2001) · Zbl 0990.20002 [3] M. Hertweck, Ganzzahlige Gruppenringe: Über den normalisator einer Gruppenbasis, preprint, 1996 [4] M. Hertweck, A solution to the isomorphism problem, preprint, 1997 [5] Jackowski, S.; Marciniak, Z., Group automorphisms inducing the identity map on cohomology, J. pure appl. algebra, 44, 1-3, 241-250, (1987) · Zbl 0624.20024 [6] Kimmerle, W., On the normalizer problem, (), 89-98 · Zbl 0931.16016 [7] Li, Y.; Parmenter, M.M.; Sehgal, S.K., On the normalizer property for integral group rings, Comm. algebra, 27, 9, 4217-4223, (1999) · Zbl 0943.16012 [8] Marciniak, Z.S.; Roggenkamp, K.W., The normalizer of a finite group in its integral group ring and cech cohomology, (), 159-188 · Zbl 0989.20002 [9] Mazur, M., Automorphisms of finite groups, Comm. algebra, 22, 6259-6271, (1994) · Zbl 0816.20019 [10] Mazur, M., On the isomorphism problem for integral group rings of infinite groups, Exposition. math., 13, 433-445, (1995) · Zbl 0841.20011 [11] Mazur, M., The normalizer of a group in the unit group of its group ring, J. algebra, 212, 1, 175-189, (1999) · Zbl 0921.16018 [12] Robinson, D.J.S., A course in the theory of groups, (1996), Springer-Verlag New York · Zbl 0496.20038 [13] Sehgal, S.K., Units in integral group rings, (1993), Longman Harlow · Zbl 0803.16022
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.