تعداد نشریات | 43 |
تعداد شمارهها | 1,650 |
تعداد مقالات | 13,402 |
تعداد مشاهده مقاله | 30,204,285 |
تعداد دریافت فایل اصل مقاله | 12,074,624 |
The Maschke property for the Sylow $p$-subgroups of the symmetric group $S_{p^n}$ | ||
International Journal of Group Theory | ||
مقاله 5، دوره 7، شماره 4، اسفند 2018، صفحه 41-64 اصل مقاله (308.75 K) | ||
نوع مقاله: Research Paper | ||
شناسه دیجیتال (DOI): 10.22108/ijgt.2017.21610 | ||
نویسندگان | ||
David J. Green1؛ L. Héthelyi2؛ E. Horváth* 3 | ||
1Institut für Mathematik Friedrich-SchillerüUniversität 07737 Jena | ||
2Budapest University of Technology and Economics, Mathematical Institute, Department of Algebra H-1111 Budapest, Műegyetem rkp. 3-9. | ||
3Budapest University of Technology and Economics, Faculty of Sciences, Inst. Math., Department of Algebra, H-1111 Budapest, Műegyetem rkp. 3-9. | ||
چکیده | ||
In this paper we prove that the Maschke property holds for coprime actions on some important classes of $p$-groups like: metacyclic $p$-groups, $p$-groups of $p$-rank two for $p>3$ and some weaker property holds in the case of regular $p$-groups. The main focus will be the case of coprime actions on the iterated wreath product $P_n$ of cyclic groups of order $p$, i.e. on Sylow $p$-subgroups of the symmetric groups $S_{p^n}$, where we also prove that a stronger form of the Maschke property holds. These results contribute to a future possible classification of all $p$-groups with the Maschke property. We apply these results to describe which normal partition subgroups of $P_n$ have a complement. In the end we also describe abelian subgroups of $P_n$ of largest size. | ||
کلیدواژهها | ||
Maschke's Theorem؛ coprime action؛ Sylow $p$-subgroup of symmetric group؛ iterated wreath product؛ uniserial action | ||
مراجع | ||
[1] Y. Berkovich, Some consequences of Maschke’s theorem, Algebra Colloq., 5 no. 2 (1998) 143–158. [2] Y. Berkovich, Groups of prime power order,. V 1, de Gruyter Expositions in Mathematics, 46 Walter de Gruyter GmbH & Co. KG, Berlin, 2008. [3] Yu. V. Bodnarchuk, Structure of the group of automorphisms of a Sylow p-subgroup of the symmetric group Spn (p ̸= 2), Ukrainian Math. J., 36 no. 6 (1984) 512–516. [4] H. Cárdenas and E. Lluis, El normalizador del p-grupo de Sylow del grupo simetrico Spn [The normalizer of the Sylow p-group of the symmetric group Spn], Bol. Soc. Mat. Mexicana, 9 (1964) 1–6. [5] S. Covello,Minimal parabolic subgroups in the symmetric groups, M. Phil. Thesis, University of Birmingham, 1998. [6] Yu. V. Dmitruk, Structure of Sylow two-subgroups of the symmetric group of degree 2n, Ukrainian Math. J., 30 no. 2 (1978) 117–124. [7] Yu. V. Dmitruk and V. I. Sushchanskii, Structure of Sylow 2-subgroups of the alternating groups and normalizers of Sylow subgroups in the symmetric and alternating groups, Ukrainian Math. J., 33 no. 3 (1981) 235–241. [8] K. Doerk and T. Hawkes, Finite soluble groups, De Gruyter Expositions in Mathematics 4, Walter de Gruyter, Berlin, New York, 1992. [9] S. Dolfi, Large orbits in coprime actions of solvable groups, Trans. Amer. Math. Soc., 360 (2008) 135–152. [10] P. Flavell, The fixed points of coprime action, Arch. Math., 75 no. 3 (2000) 173–177. [11] D. Gluck, Coprime actions with all orbit sizes small, Proc. AMS, 143 no. 6 (2015) 2371–2377. [12] D. Gorenstein, Finite groups, Chelsea Publ. Co., New York, 1980. [13] L. Héthelyi and E. Horváth, Galois actions on blocks and classes of finite groups, J. Algebra, 320 (2008) 660–679. [14] B. Huppert, Endliche Gruppen I, Springer-Verlag, Berlin, Heidelberg, New York 1967. [15] I. M. Isaacs, Finite group theory, Graduate Studies in Mathematics, 92,American Mathematical Society, Providences, Rhode Island, 2008. [16] I. M. Isaacs, M. L. Lewis and G. Navarro, Invariant characters and coprime actions on finite nilpotent groups, Arch. Math., 74 no. 6 (2000) 401–403. [17] L. Kaloujnine, La structure des p-groupes de Sylow des groupes symétriques finis, Ann. Sci.École Norm. Sup. (3), 65 (1948) 239–276. [18] H. Kurzweil and B. Stellmacher, The theory of finite groups, Springer Universitext, 2004. [19] C. R. Leedham-Green and S. McKay, The structure of groups of prime power order, London Mathematical Society Monographs, New Series, 27 Oxford University Press, Oxford, 2002. [20] Zhengxing Li, Coleman automorphisms of permutational wreath products, Comm. Alg., 44 no. 9 (2016) 3933–3938. [21] G. Malle, G. Navarro and B. Späth, Invariant blocks under coprime actions, Doc. Math., 20 (2015) 491–506. [22] V. D. Mazurov, Finite groups with metacyclic Sylow 2-subgroups, Sib. Math. J., 8 no. 5 (1967) 733–745. [23] A. Moreto and L. Sanus, Coprime actions and degrees of primitive inducers of invariant characters, Bull. Austral. Math. Soc., 64 (2001) 315–320. [24] R. C. Orellana, M. E. Orrison and D. N. Rockmore, Rooted trees and iterated wreath products, Adv. in Appl. Math., 33 (2004) 531–547. [25] J. M. Riedl, Automorphisms of regular wreath product p-groups, Int. J. Math. Math. Sci., (2009), doi:10.1155/2009/245617. [26] D. N. Rockmore, Fast Fourier transforms and wreath products, Appl. Comput. Harmon. Anal., 2 no. 3 (1995) 279–292. [27] M. Seong and A. Wu, Generalized iterated wreath products of cyclic groups and rooted tree correspondence, arXiv:1409.0603v1[math.RT] 2, 2014. [28] S. Sidki, Regular trees and their automorphisms, Monografias de Matematica, 56 , IMPA, Rio de Janeiro, 1998. [29] R. Waldecker, A theorem about coprime action, J. Algebra, 320 (2008) 2027–2030. [30] A. J. Weir, The Sylow subgroups of the symmetric groups, Proc. Amer. Math. Soc., 6 (1955) 534–541. [31] A. Woryna, The automaton realizations of iterated wreath products of cyclic groups, Comm. Alg., 42 no. 3 (2014) 1354–1361. | ||
آمار تعداد مشاهده مقاله: 1,065 تعداد دریافت فایل اصل مقاله: 875 |