Generalizing quasinormality

Document Type : Ischia Group Theory 2014


1 Australian National University

2 University of Warwick


‎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.


Main Subjects

[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.