On the normalizer problem.(English)Zbl 1063.16036

Summary: The normalizer problem of an integral group ring of an arbitrary group $$G$$ is investigated. It is shown that any element of the normalizer $${\mathcal N}_{{\mathcal U}_1}(G)$$ of $$G$$ in the group of normalized units $${\mathcal U}_1(\mathbb{Z} G)$$ is determined by a finite normal subgroup. This reduction to finite normal subgroups implies that the normalizer property holds for many classes of (infinite) groups, such as groups without non-trivial 2-torsion, torsion groups with a normal Sylow 2-subgroup, and locally nilpotent groups. Further it is shown that the commutator of $${\mathcal N}_{{\mathcal U}_1}(G)$$ equals $$G'$$ and $${\mathcal N}_{{\mathcal U}_1}(G)/G$$ is finitely generated if the torsion subgroup of the finite conjugacy group of $$G$$ is finite.

MSC:

 16U60 Units, groups of units (associative rings and algebras) 16S34 Group rings 20C07 Group rings of infinite groups and their modules (group-theoretic aspects)
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References:

 [1] Coleman, D.B., On the modular group ring of a p-group, Proc. amer. math. soc., 5, 511-514, (1964) · Zbl 0132.27501 [2] Gorenstein, D., Finite groups, (1968), Harper & Row New York · Zbl 0185.05701 [3] M. Hertweck, A counter example to the isomorphism problem for integral group rings of finite groups, preprint. · Zbl 0990.20002 [4] M. Hertweck, Eine Lösung des Isomorphieproblems für ganzzahlige Gruppenringe von endlichen Gruppen, Dissertation, Universität Stuttgart, 1998. [5] Jackowski, S.; Marciniak, Z., Group automorphisms inducing the identity map on cohomology, J. pure appl. algebra, 44, 241-250, (1997) · Zbl 0624.20024 [6] Jespers, E.; Juriaans, S.O., Isomorphisms of integral group rings of infinite groups, J. algebra, 223, 171-189, (2000) · Zbl 0947.16015 [7] Jespers, E.; Paramenter, M.M.; Sehgal, S.K., Central units of integral groups rings of nilpotent groups, Proc. amer. math. soc., 124, 1007-1012, (1996) · Zbl 0846.16028 [8] Kimmerle, W., On the normalizer problem, Algebra, Trends in mathematics, (1999), Birkhäuser Basel, p. 89-98 · Zbl 0931.16016 [9] Y. Li, The normalizer of a metabelian group in its integral group ring, preprint, 1999. [10] Li, Y., The hypercentre and the n-centre of the unit group of an integral group ring, Canad. J. math., 50, 401-411, (1998) · Zbl 0912.16013 [11] Li, Y.; Parmenter, M.M.; Sehgal, S.K., On the normalizer property for integral group rings, Comm. algebra, 27, 4217-4223, (1999) · Zbl 0943.16012 [12] Z. S. Marciniak, and, K. W. Roggenkamp, The normalizer of a finite group in its integral group ring and cech cohomology, J. Pure Appl. Algebra, in press. · Zbl 0989.20002 [13] Mazur, M., Automorphisms of finite groups, Comm. in algebra, 22, 6259-6271, (1994) · Zbl 0816.20019 [14] Mazur, M., On the isomorphism problem for infinite group rings, Exposition. math., 13, 433-445, (1995) · Zbl 0841.20011 [15] Mazur, M., The normalizer of a group in the unit group of its group ring, J. algebra, 212, 175-189, (1999) · Zbl 0921.16018 [16] T. Petit Lobão, and, C. Polcino Milies, The normalizer property for integral group rings of Frobenius groups, preprint. · Zbl 1017.16024 [17] Polcino Milies, C.; Sehgal, S.K., Central units of integral group rings, Comm. algebra, 27, 6233-6241, (1999) · Zbl 0943.16014 [18] Robinson, D.J.S., A course in the theory of groups, (1980), Springer-Verlag New York/Heidelberg · Zbl 0496.20038 [19] Sehgal, S.K., Topics in group rings, (1978), Dekker New York · Zbl 0411.16004 [20] Sehgal, S.K., Units in integral group rings, (1993), Longman Essex · Zbl 0803.16022
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