On some invariants of finite groups

Document Type : Ischia Group Theory 2012


1 Institute of Mathematics, University of Warsaw

2 Institute of Mathematics University of Białystok


A normal subgroup $N$ of a group $G$ is said to be an‎ omissible subgroup of $G$ if it has the following property‎: ‎whenever $X\leq G$ is such that $G=XN$‎, ‎then $G=X$‎. ‎In this note we construct various groups $G$‎, ‎each of which has an omissible subgroup $N\neq 1$ such that $G/N\cong SL_2(k)$ where $k$ is a field of positive characteristic‎.


Main Subjects

A. Aljouiee and F. Alrusaini (2010). Matroid groups and basis property. Int. J. Algebra. 4, 535-540 Y. Berkovich (2008). Groups of Prime Power Order. Walter de Gruyter, Berlin. 1 H. U. Besche, B. Eick and E. A. O'Brien (2002). A millenium project: constructing small groups. Internat. J. Algebra Comput.. 12, 623-644 E. Crestani and F. Menegazzo (2012). On monotone $2$-groups. J. Group Theory. 15, 359-383 D. Gorenstein (1980). Finite Groups. Chelsea Publishing Company, New York. G. Gr"{a}tzer (1998). General Lattice Theory. Birkh"auser Verlag, Basel. P. Grzeszczuk and E. R. Puczyl{}owski (1995). A radical of lattices and its applications to rings and modules. Contributions to General Algebra. 9, 203-212 B. Huppert (1983). Endliche Gruppen I. Springer-Verlag, Berlin. P. R. Jones (1978). Basis properties for inverse semigroups. J. Algebra. 50, 135-152 G. Malle, J. Saxl and T. Weigel (1994). Generation of classical groups. Geom. Dedicata. 49, 85-116 A. Mann (2005). The number of generators of finite $p$-groups. J. Group Theory. 8, 317-337 J. McDougall-Bagnall and M. Quick (2011). Groups with the basis property. J. Algebra. 346, 332-339 W. Narkiewicz (1983). Number Theory. World Scientific Publishing Co., Singapore. D. J. S. Robinson (1995). A Course in the Theory of Groups. Springer-Verlag, New York. R. Scapellato and L. Verardi (1994). Bases of certain finite groups. Ann. Math. Blaise Pascal. 1, 85-93 R. Schmidt (1994). Subgroup Lattices of Groups. Walter de Gruyter, Berlin. J. Whiston (2000). On the maximal size of independent generating sets of the symmetric group. J. Algebra. 232, 255-268