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Coleman automorphisms of finite groups with a self-centralizing normal subgroup. (English) Zbl 07285991
Summary: Let $$G$$ be a finite group with a normal subgroup $$N$$ such that $$C_{G}(N)\leq N$$. It is shown that under some conditions, Coleman automorphisms of $$G$$ are inner. Interest in such automorphisms arose from the study of the normalizer problem for integral group rings.
##### MSC:
 20C05 Group rings of finite groups and their modules (group-theoretic aspects) 16S34 Group rings 20C10 Integral representations of finite groups
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##### References:
 [1] Coleman, D. B., On the modular group ring of a $$p$$-group, Proc. Am. Math. Soc. 15 (1964), 511-514 [2] Hai, J.; Ge, S.; He, W., The normalizer property for integral group rings of holomorphs of finite nilpotent groups and the symmetric groups, J. Algebra Appl. 16 (2017), Article ID 1750025, 11 pages [3] Hai, J.; Guo, J., The normalizer property for integral group ring of the wreath product of two symmetric groups $$S_k$$ and $$S_n$$, Commun. Algebra 45 (2017), 1278-1283 [4] Hai, J.; Li, Z., On class-preserving Coleman automorphisms of finite separable groups, J. Algebra Appl. 13 (2014), Article ID 1350110, 8 pages [5] Hertweck, M., A counterexample to the isomorphism problem for integral group rings, Ann. Math. (2) 154 (2001), 115-138 [6] Hertweck, M., Local analysis of the normalizer problem, J. Pure Appl. Algebra 163 (2001), 259-276 [7] Hertweck, M.; Jespers, E., Class-preserving automorphisms and the normalizer property for Blackburn groups, J. Group Theory 12 (2009), 157-169 [8] Hertweck, M.; Kimmerle, W., Coleman automorphisms of finite groups, Math. Z. 242 (2002), 203-215 [9] Huppert, B., Endliche Gruppen I, Grundlehren der mathematischen Wissenschaften 134. Springer, Berlin (1967), German [10] Jackowski, S.; Marciniak, Z., Group automorphisms inducing the identity map on cohomology, J. Pure Appl. Algebra 44 (1987), 241-250 [11] Jespers, E.; Juriaans, S. O.; Miranda, J. M. de; Rogerio, J. R., On the normalizer problem, J. Algebra 247 (2002), 24-36 [12] Juriaans, S. O.; Miranda, J. M. de; Robério, J. R., Automorphisms of finite groups, Commun. Algebra 32 (2004), 1705-1714 [13] Li, Y., The normalizer of a metabelian group in its integral group ring, J. Algebra 256 (2002), 343-351 [14] Marciniak, Z. S.; Roggenkamp, K. W., The normalizer of a finite group in its integral group ring and Čech cohomology, Algebra - Representation Theory NATO Sci. Ser. II Math. Phys. Chem. 28. Kluwer Academic, Dordrecht (2001), 159-188 [15] Lobão, T. Petit; Milies, C. Polcino, The normalizer property for integral group rings of Frobenius groups, J. Algebra 256 (2002), 1-6 [16] Lobão, T. Petit; Sehgal, S. K., The normalizer property for integral group rings of complete monomial groups, Commun. Algebra 31 (2003), 2971-2983 [17] Rose, J. S., A Course on Group Theory, Cambridge University Press, Cambridge (1978) [18] Sehgal, S. K., Units in Integral Group Rings, Pitman Monographs and Surveys in Pure and Applied Mathematics 69. Longman Scientific & Technical, Harlow (1993) [19] Antwerpen, A. Van, Coleman automorphisms of finite groups and their minimal normal subgroups, J. Pure Appl. Algebra 222 (2018), 3379-3394
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