Abstract
We study several well-known questions on pronormality and strong pronormality of Hall subgroups. In particular, we exhibit the examples of finite groups (a) having a Hall subgroup not pronormal in its normal closure (this solves Problem 18.32 of The Kourovka Notebook in the negative); (b) having a Hall subgroup pronormal but not strongly pronormal; and (c) that are simple, having a Hall subgroup, and not strongly pronormal (this solves Problem 17.45(b) of The Kourovka Notebook in the negative).
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References
Vdovin E. P. and Revin D. O., “Theorems of Sylow type,” Russian Math. Surveys, vol. 66, no. 5, 829–870 (2011).
Vdovin E. P. and Revin D. O., “The existence of pronormal π-Hall subgroups in Eπ-groups,” Sib. Math. J., vol. 56, no. 3, 379–383 (2015).
Vdovin E. P. and Revin D. O., “On the pronormality of Hall subgroups,” Sib. Math. J., vol. 54, no. 1, 22–28 (2013).
Guo W. and Revin D. O., “On the class of groups with pronormal Hall p-subgroups,” Sib. Math. J., vol. 55, no. 3, 415–427 (2014).
Vdovin E. P. and Revin D. O., “Pronormality of Hall subgroups in finite simple groups,” Sib. Math. J., vol. 53, no. 3, 419–430 (2012).
Vdovin E. P. and Revin D. O., “Pronormality and strong pronormality of subgroups,” Algebra and Logic, vol. 52, no. 1, 15–23 (2013).
Nesterov M. N., “Pronormality of Hall subgroups in almost simple groups,” Sib. Elektron. Mat. Izv., vol. 12, 1032–1038 (2015).
Manzaeva N. C., “On the heritability of the Hall property Dp by overgroups of p-Hall subgroups,” arXiv:1504.03137 (2015).
Vdovin E. P. and Revin D. O., “A conjugacy criterion for Hall subgroups in finite groups,” Sib. Math. J., vol. 51, no. 3, 402–409 (2010).
Revin D. O. and Vdovin E. P., “Frattini argument for Hall subgroups,” J. Algebra, vol. 414, 95–104 (2014).
Mazurov V. D. and Khukhro E. I. (Eds.), The Kourovka Notebook: Unsolved Problems in Group Theory. 18th ed., Sobolev Inst. Math., Novosibirsk (2014).
Nesterov M. N., “Arithmetic of conjugacy of p-complements,” Algebra and Logic, vol. 54, no. 1, 36–47 (2015).
Ljunggren W., “Some theorem on indeterminate equations of the form x n-1/x-1 = y q,” Norsk Mat. Tidsskr., vol. 25, 17–20 (1943).
Bugeaud Y. and Mihailescu P., “On the Nagell–Ljunggren equation x n-1/x-1 = y q,” Math. Scand., vol. 101, no. 2, 177–183 (2007).
Aschbacher M., Finite Group Theory, Cambridge Univ. Press, Cambridge (1986).
Kleidman P. B. and Liebeck M. W., The Subgroup Structure of Finite Classical Groups, Cambridge Univ. Press, Cambridge (1990).
Conway J. H., Curtis R. T., Norton S. P., Parker R. A., and Wilson R. A., Atlas of Finite Groups, Clarendon Press, Oxford (1985).
Hall P., “Theorems like Sylow’s,” Proc. London Math. Soc., vol. 3, no. 2, 286–304 (1956).
Doerk K. and Hawkes T. O., Finite Soluble Groups, Walter de Gruyter, Berlin and New York (1992).
Gross F., “Conjugacy of odd order Hall subgroups,” Bull. London Math. Soc., vol. 19, no. 4, 311–319 (1987).
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The author was supported by the Russian Science Foundation (Grant 14–21–00065).
Novosibirsk. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 58, No. 1, pp. 165–173, January–February, 2017; DOI: 10.17377/smzh.2017.58.116.
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Nesterov, M.N. On pronormality and strong pronormality of Hall subgroups. Sib Math J 58, 128–133 (2017). https://doi.org/10.1134/S0037446617010165
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DOI: https://doi.org/10.1134/S0037446617010165