On finite factorized groups with permutable subgroups of factors
- PDF / 326,221 Bytes
- 9 Pages / 439.37 x 666.142 pts Page_size
- 100 Downloads / 212 Views
Archiv der Mathematik
On finite factorized groups with permutable subgroups of factors Victor S. Monakhov and Alexander A. Trofimuk
Abstract. Two subgroups A and B of a group G are called msp-permutable if the following statements hold: AB is a subgroup of G; the subgroups P and Q are mutually permutable, where P is an arbitrary Sylow psubgroup of A and Q is an arbitrary Sylow q-subgroup of B, p = q. In the present paper, we investigate groups that are factorized by two msppermutable subgroups. In particular, the supersolubility of the product of two supersoluble msp-permutable subgroups is proved. Mathematics Subject Classification. 20D10, 20D20. Keywords. Mutually permutable subgroups, Sylow subgroups, msp-Permutable subgroups, Supersoluble groups.
1. Introduction. Throughout this paper, all groups are finite and G always denotes a finite group. We use the standard notations and terminology of [3]. The notation Y ≤ X means that Y is a subgroup of a group X. Two subgroups A and B of a group G are called mutually (totally) permutable if U B = BU and AV = V A (respectively, U V = V U ) for all U ≤ A and V ≤ B. The idea of totally and mutually permutable subgroups was first initiated by Asaad and Shaalan in [1]. This direction has since been subject of an indepth study of many authors. An exhaustive report on this matter appears in [3, chapters 4–5]. It is quite natural to consider a factorized group G = AB in which certain subgroups of the factors A and B are mutually (totally) permutable. In this direction, Monakhov [7] obtained the solubility of a group G = AB under the assumption that the subgroups A and B are soluble and the Carter subgroups (Sylow subgroups) of A and of B are permutable. We introduce the following
V.S. Monakhov and A.A. Trofimuk
Arch. Math.
Definition. Two subgroups A and B of a group G are called msp-permutable if the following statements hold: (1) AB is a subgroup of G; (2) the subgroups P and Q are mutually permutable, where P is an arbitrary Sylow p-subgroup of A and Q is an arbitrary Sylow q-subgroup of B, p = q. In the present paper, we investigate groups that are factorized by two msppermutable subgroups. In particular, the supersolubility of the product of two supersoluble msp-permutable subgroups is proved. 2. Preliminaries. In this section, we give some definitions and basic results which are essential in the sequel. A group whose chief factors have prime orders is called supersoluble. Recall that a p-closed group is a group with a normal Sylow p-subgroup and a p-nilpotent group is a group with a normal Hall p -subgroup. Denote by G , Z(G), F (G), and Φ(G) the derived subgroup, centre, Fitting, and Frattini subgroups of G, respectively; P the set of all primes. We use Ept to denote an elementary abelian group of order pt and Zm to denote a cyclic group of order m. The semidirect product of a normal subgroup A and a subgroup B is written as follows: A B. The monographs [2,5] contain the necessary information of the theory of formations. The formations of all nilpotent, p-group
Data Loading...
