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Local analysis of the normalizer problem. (English) Zbl 0987.16015
Let $$G$$ be a finite group and let $$RG$$ be the group ring of $$G$$ over the commutative ring $$R$$. In the paper under review the author studies various subgroups of the group $$\text{Out}(G)$$ of outer automorphism of $$G$$. An essential ingredient of the author’s spectacular counterexample to the isomorphism problem for integral group rings was the existence of a non-trivial element in the kernel $$\text{Out}_R(G)$$ of $$\text{Out}(G)\to\text{Out}(RG)$$ for $$R=\mathbb{Z}$$. The author proves a variety of results related to this group in an extremely elegant way. We only cite some of them. Noting by $$A$$ the ring of algebraic integers in the complex numbers, he reviews an earlier paper of K. W. Roggenkamp and the reviewer [J. Pure Appl. Algebra 103, No. 1, 91-99 (1995; Zbl 0835.16020)] where an example of a group with $$\text{Out}_A(G)\neq 1$$ was given. The author simplifies the presentation which was not optimal there and modifies the above group to obtain a group $$G$$ of order 3200 with $$\text{Out}_A(G)\neq 1$$. Using this improved method, the author proves that $$\text{Out}_A(G)$$ is Abelian, and any finite Abelian group can occur for metablian $$G$$. Moreover, using a modification of an argument of Krempa, he proves that $$\text{Out}_A(G)$$ is in the centre of the group of class preserving automorphisms. Finally, the author studies the case of groups with Abelian Sylow 2 subgroups. He produces a group $$G$$ with all Sylow subgroups Abelian and $$\text{Out}_A(G)\neq 1=\text{Out}_\mathbb{Z}(G)$$.

##### MSC:
 16S34 Group rings 20C10 Integral representations of finite groups 20F28 Automorphism groups of groups 20F29 Representations of groups as automorphism groups of algebraic systems 20C05 Group rings of finite groups and their modules (group-theoretic aspects) 16W20 Automorphisms and endomorphisms
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##### References:
 [1] Aschbacher, M., Finite group theory, (1986), Cambridge University Press Cambridge · Zbl 0583.20001 [2] Clifford, A.H., Representations induced in an invariant subgroup, Ann. math., 38, 533-550, (1937) · Zbl 0017.29705 [3] Coleman, D.B., On the modular group ring of a p-group, Proc. amer. math. soc., 15, 511-514, (1964) · Zbl 0132.27501 [4] Curtis, C.W.; Reiner, I., Methods of representation theory, vols. I, II, (1981, 1987), Wiley New York [5] Dade, E.C., Locally trivial outer automorphisms of finite groups, Math. Z., 114, 173-179, (1970) · Zbl 0194.33601 [6] Fröhlich, A., The Picard groups of non-commutative rings, in particular of orders, Trans. amer. math. soc., 180, 1-46, (1973) · Zbl 0278.16016 [7] M. Hertweck, Zentrale und primzentrale Automorphismen, Darstellungstheorietage 7-9 Mai 92 in Erfurt, Sitzungsber. Math.-Naturwiss. Kl., No. 4, 1992, pp. 67-76. · Zbl 0810.20017 [8] M. Hertweck, Eine Lösung des Isomorphieproblems für ganzzahlige Gruppenringe von endlichen Gruppen, Ph.D. Thesis, University of Stuttgart, 1998. [9] M. Hertweck, A solution of the isomorphism problem for integral group rings, manuscript, 1999, submitted for publication. [10] M. Hertweck, Class-preserving automorphisms of finite groups, manuscript, 1999, submitted for publication. [11] Huppert, B., Endliche gruppen I, (1967), Springer Berlin · Zbl 0217.07201 [12] Jackowski, S.; Marciniak, Z., Group automorphisms inducing the identity map on cohomology, J. pure appl. algebra, 44, 241-250, (1987) · Zbl 0624.20024 [13] Jacobinski, H., Über die geschlechter von gittern über ordnungen, J. reine angew. math., 230, 29-39, (1968) · Zbl 0157.10403 [14] Kimmerle, W., On the normalizer problem, (), 89-98 · Zbl 0931.16016 [15] Kimmerle, W.; Roggenkamp, K.W., Projective limits of group rings, J. pure appl. algebra, 88, 119-142, (1993) · Zbl 0786.20002 [16] Li, Y., On the normalizers of the unitary subgroup in an integral group ring, Comm. algebra, 25, 10, 3267-3282, (1997) · Zbl 0884.16019 [17] Mazur, M., Automorphisms of finite groups, Comm. algebra, 22, 15, 6259-6271, (1994) · Zbl 0816.20019 [18] M. Mazur, On the isomorphism problem for infinite group rings, Expositiones Mathematicae, vol. 13, Spektrum Akademischer Verlag, Heidelberg, 1995, pp. 433-445. · Zbl 0841.20011 [19] 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 [20] Passman, D.S., The algebraic structure of group rings, (1977), Wiley-Interscience New York · Zbl 0366.16003 [21] Roggenkamp, K.W., An extension of the noether – deuring theorem, Proc. amer. math. soc., 31, 423-426, (1972) · Zbl 0258.16029 [22] K.W. Roggenkamp, The isomorphism problem for integral group rings of finite groups, Progress in Mathematics, vol. 95, Birkhäuser, Basel, 1991, pp. 193-220. · Zbl 0752.20003 [23] K.W. Roggenkamp, Group rings: units and the isomorphism problem, in: K.W. Roggenkamp, M.J. Taylor, Group Rings and Class Groups, DMV Seminar Band 18, Birkhäuser, Basel, 1992, Part I. · Zbl 0769.20004 [24] Roggenkamp, K.W.; Taylor, M.J., Group rings and class groups, DMV seminar band 18, (1992), Birkhäuser Basel [25] Roggenkamp, K.W.; Zimmermann, A., Outer group automorphisms may become inner in the integral group ring, J. pure appl. algebra, 103, 91-99, (1995) · Zbl 0835.16020 [26] A.I. Saksonov, On the group ring of finite groups I, Publ. Math. Debrecen 18 (1971) 187-209 (in Russian). · Zbl 0256.20007 [27] L.L. Scott, Recent progress on the isomorphism problem, in: Representations of Finite Groups, Proceedings Conference, Arcata, CA, 1986, Proceedings of Symposium on Pure Mathematics, vol. 47, 1987, pp. 259-274. [28] S.K. Sehgal, Units in Integral Group Rings, Pitman Monographs Surveys Pure and Applied Mathematics, vol. 96, Longman Scientific & Technical, Essex, 1993. · Zbl 0803.16022
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