zbMATH — the first resource for mathematics

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.
20C05 Group rings of finite groups and their modules (group-theoretic aspects)
16S34 Group rings
20C10 Integral representations of finite groups
Full Text: DOI
[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
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.