Influence ofS-quasinormality condition on almost minimal subgroups of a finite group

Abstract

Let r be a positive rational integer, and G a finite group. A subgroup H of G is called a r minimal (r-maximal) subgroup if there exists a subgroup chain 1 = H o ~< H 1 ~< -.. ~<H, = H(H = Ho ~< Hx ~<'" ~< H, = G), where H i is maximal in Hi+ x, we call Han S-quasinormal subgroup of G, if HP = PH for any Sylow subgroup P of G. If H is S-quasinormal in G, we write H SQN G. A subgroup H of G is called a quasinormal subgroup if HK = KH for any subgroup K of G. We write H QN G when H is a quasinormal subgroup of G. G is said to be an SQN-r-group(QN-r-group, PN-r-group) if every r-minimal subgroup of G is S-quasinormal(quasinormal, normal) in G. Clearly, a PN-r-group is a ON-r-group, and a QN-r-group is an SQN-r-group. Gaschfitz and Ito proved that PNl -g roup G is solvable and its commutator subgroup G' isp-nilpotent for any odd prime p. Hence the Fitting length Fl(G) ~< 3 [5, th5.7 p436]. Buckley proved in [1] that PN-1group of odd order is supersolvable. Sastry and Deskins had studied in I'3] PN-r-group and QNr-group. For instance, they proved that QN-r-group (r = 1,2,3) is solvable and QN-l-group and QN-2-group have Fitting length at most 4, the Fitting length of QN-3-group is not more than 5. This paper will study the SQN-r-groups. In section 2, we explore the properties of S--quasinormal subgroup and prove that the normal Hall subgroups of an S-quasinormal subgroup of a finite group is also an S-quasinormal subgroup. In section 3, we prove that SQN-l-group is solvable and the nitpotent residual is p-nilpotent for any odd prime p, so the Fitting length is at most 3. In section 4, we prove that SQN-2-group and SQN-3-group are either supersolvable or isomorphic to the several classes groups given in section t. Moreover, the commutator subgroups are nilpotent, so the Fitting lengths are at most 2. In section 5, we show that SQN-4-group and SQN-5--group are either solvable and the Fitting lengths at most 3 or non-solvable with only one non-abelian principal factor ~r is isomorphic ~o the simple group PSL(2,p). Groups in this .paper are all of finite order. The author wishes to thank his supervisor Professor Duan Xuefu and-Xu Mingyiao for their encouragement and many instructions.