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Generalizing quasinormality | ||
| International Journal of Group Theory | ||
| مقاله 6، دوره 4، شماره 1، خرداد 2015، صفحه 33-39 اصل مقاله (190.47 K) | ||
| نوع مقاله: Ischia Group Theory 2014 | ||
| شناسه دیجیتال (DOI): 10.22108/ijgt.2015.7326 | ||
| نویسندگان | ||
| John Cossey1؛ Stewart Edward Stonehewer* 2 | ||
| 1Australian National University | ||
| 2University of Warwick | ||
| چکیده | ||
| Quasinormal subgroups have been studied for nearly 80 years. In finite groups, questions concerning them invariably reduce to $p$-groups, and here they have the added interest of being invariant under projectivities, unlike normal subgroups. However, it has been shown recently that certain groups, constructed by Berger and Gross in 1982, of an important universal nature with regard to the existence of core-free quasinormal subgroups generally, have remarkably few such subgroups. Therefore in order to overcome this misfortune, a generalization of the concept of quasinormality will be defined. It could be the beginning of a lengthy undertaking. But some of the initial findings are encouraging, in particular the fact that this larger class of subgroups also remains invariant under projectivities of finite $p$-groups, thus connecting group and subgroup lattice structures. | ||
| کلیدواژهها | ||
| p-groups؛ quasinormal؛ products | ||
| مراجع | ||
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[1] T. R. Berger and F. Gross, A universal example of a core-free permutable subgroup, Rocky Mountain J. Math., 12 (1982) 345-365.
[2] J. Cossey and S. E. Stonehewer, On the rarity of quasinormal subgroups, Rend. Semin. Mat. Univ. Padova, 125 (2011) 81-105.
[3] B. Huppert, Endliche Gruppen. I, Springer-Verlag, Berlin Heidelberg, New York, 1967.
[4] N. Itô and J. Szép,Über die Quasinormalteiler von endlichen Gruppen, Acta Sci. Math. (Szeged), 23 (1962) 168-170.
[5] 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, 2002.
[6] R. Maier and P. Schmid, The embedding of permutable subgroups in finite groups, Math. Z., 131 (1973) 269-272.
[7] O. Ore, Structures and group theory. I, Duke Math., 3 (1937) 149-173.
[8] R. Schmidt, Subgroup Lattices of Groups, Expositions in Mathematics, 14, de Gruyter, Berlin, New York, 1994.
[9] S. E. Stonehewer and G. Zacher, Cyclic Quasinormal Subgroups of Arbitrary Groups, Rend. Sem. Mat. Univ. Padova, 115 (2006) 165-187. [10] J. G. Thompson, An example of core-free quasinormal subgroups of p-groups, Math. Z., 96 (1967) 226-227. | ||
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