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Keywords:
Coleman automorphism; integral group ring; the normalizer property
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.\looseness -1
References:
[1] Coleman, D. B.: On the modular group ring of a $p$-group. Proc. Am. Math. Soc. 15 (1964), 511-514. DOI 10.1090/S0002-9939-1964-0165015-8 | MR 0165015 | Zbl 0132.27501
[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. DOI 10.1142/S0219498817500256 | MR 3608412 | Zbl 1388.20008
[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. DOI 10.1080/00927872.2016.1175613 | MR 3573379 | Zbl 1372.20007
[4] Hai, J., Li, Z.: On class-preserving Coleman automorphisms of finite separable groups. J. Algebra Appl. 13 (2014), Article ID 1350110, 8 pages. DOI 10.1142/S0219498813501107 | MR 3125878 | Zbl 1302.20028
[5] Hertweck, M.: A counterexample to the isomorphism problem for integral group rings. Ann. Math. (2) 154 (2001), 115-138. DOI 10.2307/3062112 | MR 1847590 | Zbl 0990.20002
[6] Hertweck, M.: Local analysis of the normalizer problem. J. Pure Appl. Algebra 163 (2001), 259-276. DOI 10.1016/S0022-4049(00)00167-5 | MR 1852119 | Zbl 0987.16015
[7] Hertweck, M., Jespers, E.: Class-preserving automorphisms and the normalizer property for Blackburn groups. J. Group Theory 12 (2009), 157-169. DOI 10.1515/JGT.2008.068 | MR 2488146 | Zbl 1168.16017
[8] Hertweck, M., Kimmerle, W.: Coleman automorphisms of finite groups. Math. Z. 242 (2002), 203-215. DOI 10.1007/s002090100318 | MR 1980619 | Zbl 1047.20020
[9] Huppert, B.: Endliche Gruppen I. Grundlehren der mathematischen Wissenschaften 134. Springer, Berlin (1967), German. DOI 10.1007/978-3-642-64981-3 | MR 0224703 | Zbl 0217.07201
[10] Jackowski, S., Marciniak, Z.: Group automorphisms inducing the identity map on cohomology. J. Pure Appl. Algebra 44 (1987), 241-250. DOI 10.1016/0022-4049(87)90028-4 | MR 0885108 | Zbl 0624.20024
[11] Jespers, E., Juriaans, S. O., Miranda, J. M. de, Rogerio, J. R.: On the normalizer problem. J. Algebra 247 (2002), 24-36. DOI 10.1006/jabr.2001.8724 | MR 1873381 | Zbl 1063.16036
[12] Juriaans, S. O., Miranda, J. M. de, Robério, J. R.: Automorphisms of finite groups. Commun. Algebra 32 (2004), 1705-1714. DOI 10.1081/AGB-120029897 | MR 2099696 | Zbl 1072.20030
[13] Li, Y.: The normalizer of a metabelian group in its integral group ring. J. Algebra 256 (2002), 343-351. DOI 10.1016/S0021-8693(02)00102-3 | MR 1939109 | Zbl 1017.16023
[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. DOI 10.1007/978-94-010-0814-3_8 | MR 1858036 | Zbl 0989.20002
[15] Lobão, T. Petit, Milies, C. Polcino: The normalizer property for integral group rings of Frobenius groups. J. Algebra 256 (2002), 1-6. DOI 10.1016/S0021-8693(02)00156-4 | MR 1936875 | Zbl 1017.16024
[16] Lobão, T. Petit, Sehgal, S. K.: The normalizer property for integral group rings of complete monomial groups. Commun. Algebra 31 (2003), 2971-2983. DOI 10.1081/AGB-120021903 | MR 1986226 | Zbl 1039.16034
[17] Rose, J. S.: A Course on Group Theory. Cambridge University Press, Cambridge (1978). MR 0498810 | Zbl 0371.20001
[18] Sehgal, S. K.: Units in Integral Group Rings. Pitman Monographs and Surveys in Pure and Applied Mathematics 69. Longman Scientific & Technical, Harlow (1993). MR 1242557 | Zbl 0803.16022
[19] Antwerpen, A. Van: Coleman automorphisms of finite groups and their minimal normal subgroups. J. Pure Appl. Algebra 222 (2018), 3379-3394. DOI 10.1016/j.jpaa.2017.12.013 | MR 3806731 | Zbl 06881278
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