Weak closure and Oliver's p-group conjecture

To date almost all verifications of Oliver's p-group conjecture have proceeded by verifying a stronger conjecture about weakly closed quadratic subgroups. We construct a group of order 3^n for n = 49 which refutes the weakly closed conjecture but satisfies Oliver's conjecture.Comment: 9 page

On the Sylow graph of a finite group

The final publication is available at Springer via http://dx.doi.org/10.1007/s11856-011-0138-xLet G be a finite group and Gp be a Sylow p-subgroup of G for a prime p in pi(G), the set of all prime divisors of the order of G. The automiser Ap(G) is defined to be the group NG(Gp)/GpCG(Gp). We define the Sylow graph gamma A(G) of the group G, with set of vertices pi(G), as follows: Two vertices p, q ¿ ¿(G) form an edge of ¿A(G) if either q ¿ ¿(Ap(G)) or p ¿ ¿(Aq(G)). The following result is obtained: Theorem: Let G be a finite almost simple group. Then the graph ¿A(G) is connected and has diameter at most 5. We also show how this result can be applied to derive information on the structure of a group from the normalizers of its Sylow subgroups.The second and third authors have been supported by Proyecto MTM2007-68010-C03-03 and Proyecto MTM2010-19938-C03-02, Ministerio de Educacion y Ciencia and FEDER, Spain.Kazarin, SL.; Martínez Pastor, A.; Pérez-Ramos, M. (2011). On the Sylow graph of a finite group. Israel Journal of Mathematics. 186(1):251-271. doi:10.1007/s11856-011-0138-xS2512711861Z. Arad and D. Chillag, Finite groups containing a nilpotent Hall subgroup of even order, Houston Journal of Mathematics 7 (1981), 23–32.H. Azad, Semi-simple elements of order 3 in finite Chevalley groups, Journal of Algebra 56 (1979), 481–498.A. Ballester-Bolinches, A. Martínez-Pastor, M. C. Pedraza-Aguilera and M. D. Pérez-Ramos, On nilpotent-like fitting formations, in Groups St. Andrews 2001 in Oxford, (C. M. Campbell et al., eds.) London Mathematical Society Lecture Note Series 304, Cambridge University Press, 2003, pp. 31–38.M. Bianchi, A. Gillio Berta Mauri and P. Hauck, On finite groups with nilpotent Sylow normalizers, Archiv der Mathematik 47 (1986), 193–197.A. Borel, R. Carter, C.W. Curtis, N. Iwahori, T. A. Springer, R. Steinberg, Seminar on Algebraic Groups and Related Finite Groups, Lecture Notes of Mathematics 131 Springer, Berlin, 1970.N. Bourbaki, Éléments de mathématique: Groupes et algèbres de Lie, Chapters IV, V, VI, Hermann, Paris, 1968.R. W. Carter, Simple groups of Lie type, Wiley, London, 1972.R. W. Carter, Conjugacy classes in the Weyl group, Compositio Mathematica 25 (1972), 1–59.R. W. Carter, Finite Groups of Lie Type: Conjugacy Classes and Complex Characters, Wiley, London, 1985.A. D’Aniello, C. De Vivo and G. Giordano, On certain saturated formations of finite groups, in Proceedings Ischia Group Theory 2006, (T. Hawkes, P. Longobardy and M. Maj, eds.) World Scientific, Hackensack, NJ, 2007, pp. 22–32.A. D’Aniello, C. De Vivo and G. Giordano, Lattice formations and Sylow normalizers: a conjecture, Atti del Seminario Matematico e Fisico dell’ Università di Modena e Reggio Emilia 55 (2007), 107–112.A. D’Aniello, C. De Vivo, G. Giordano and M. D. Pérez-Ramos, Saturated formations closed under Sylow normalizers, Communications in Algebra 33 (2005), 2801–2808.K. Doerk, T. Hawkes, Finite soluble groups, Walter De Gruyter, Berlin-New York, 1992.G. Glauberman, Prime-power factor groups of finite groups II, Mathematische Zeitschrift 117 (1970), 46–56.D. Gorenstein, R. Lyons, The local 2-structure of groups of characteristic 2 type, Memoirs of the American Mathematical Society 42, No. 276, Providence, RI, 1983.R. M. Guralnick, G. Malle and G. Navarro, Self-normalizing Sylow subgroups, Proceedings of the American Mathematical Society 132 (2004), 973–979.F. Menegazzo, M. C. Tamburini, A property of Sylow p-normalizers in simple groups, Quaderni del seminario Matematico di Brescia, n. 45/02 (2002).R. Steinberg, Lectures on Chevalley Groups, Yale University, New Haven, Conn., 1968.E. Stensholt, An application of Steinberg’s construction of twisted groups, Pacific Journal of Mathematics 55 (1974), 595–618.E. Stensholt, Certain embeddings among finite groups of Lie type, Journal of Algebra 53 (1978), 136–187.K. Zsigmondy, Zur Theorie der Potenzreste, Monatshefte für Mathematik and Physik 3 (1892), 265–284