Factorizable Groups and Formations
- PDF / 229,130 Bytes
- 8 Pages / 595 x 842 pts (A4) Page_size
- 72 Downloads / 224 Views
© Springer 2005
Factorizable Groups and Formations KH. A. AL-SHARO1 , E. A. MOLOKOVA2 and L. A. SHEMETKOV3 1 Al al-Bayt University, Mafraq, Jordan 2 Gomel State Technical University, Belarus 3 F. Scorina Gomel State University, Belarus. e-mail: [email protected]
Abstract. We give two characterizations of saturated formations of finite groups in terms of complemented and supplemented subgroups. Mathematics Subject Classifications (2000): 20D10, 20D40. Key words: finite group, formations.
1. Introduction Let a group G be a product of two its subgroups A and B, i.e. G = AB. Then B is called a supplement to A in G. If A ∩ B = 1, then B is called a complement to A in B. Completely factorizable groups (i.e. groups in which every subgroup is complemented) in the universe of finite groups were investigated in [8], and in the general case they were investigated in [3, 4]. In particular, finite completely factorizable groups are contained in the formation U of all supersoluble finite groups. Since 1953, S. N. Chernikov and his disciples have carried out a series of works investigating the groups with a prescribed system of complemented subgroups. In many studied cases (see [6], Chapter 7-8 and Addendum), the groups either turned out to be supersoluble or had properties similar to the properties of U-groups. We mention the following Chernikov’s theorem as an example. THEOREM 1.1 ([5]). Let G be a finite group such that every Sylow subgroup of G is Abelian. The group G is supersoluble if and only if every primary cyclic subgroup complemented in a Sylow subgroup of G is complemented in G. We observe a similar situation while studying the groups with a system of supplemented subgroups. We say that a subgroup A is strictly supplemented in G if there exists a supplement B = G to A in G. V. A. Vedernikov and N. I. Kuleshov proved the following theorem. THEOREM 1.2 ([16]). A finite group G is supersoluble if and only if every primary cyclic subgroup strictly supplemented in a Sylow subgroup of G is strictly supplemented in G.
4
KH. A. AL-SHARO ET AL.
In [1] a notion of ZF,G -element was introduced for a saturated formation F. In fact, in Theorems 1.1–1.2, the generators of the considered supplemented cyclic subgroups are ZU,G -elements, and the authors proved that under the considered assumptions, G is supersoluble. In the present article, we extend the mentioned theorems considering instead, U to be an arbitrary saturated formation. All the groups considered are finite. 2. Preliminaries Let F be a formation of groups, i.e. F is a group class closed by taking homomorphic images and subdirect products. If F is a nonempty formation, then the F-residual GF of G is the smallest normal subgroup of G with a quotient in F. A formation F is called saturated if G/(G) ∈ F always implies G ∈ F. A function f : {primes} → {formations} is called a local satellite. A chief factor H /K of G is called f -central if G/CG (H /K) ∈ f (p) for every prime divisor p of |H /K|. We denote by LF(f ) the class of all the groups whose chief factor
Data Loading...
