Siberian Mathematical Journal, Vol. 48, No. 1, pp. 150–155, 2007 Original Russian Text Copyright c  2007 Safonov V. G. CHARACTERIZATION OF THE SOLUBLE ONE-GENERATED TOTALLY SATURATED FORMATIONS OF FINITE GROUPS V. G. Safonov UDC 512.542 Abstract: We prove that the finite length condition for the lattice of τ -closed totally saturated sub- formations of a τ -closed totally saturated formation is equivalent to the finite length condition for this lattice, and it is also equivalent for the formation to be soluble and one-generated. Keywords: formation of finite groups, totally saturated formation, lattice of formations, τ -closed formation, one-generated formation 1. Introduction In the formation theory of finite groups, the following problem is well known: Is the number finite of the subformations of a one-generated formation form G (see [1, Problem 2, p. 18; 2, Question 9.59; 3, p. 482])? An analogous problem for an n-multiply saturated one-generated formation l n form G is also open [4, Problem 3.51, p. 49; 5, Problem 22, p. 218]. In [6], R. M. Bryant, R. A. Bryce, and B. Hartley showed that, for every soluble group G, the forma- tion form G (l form G) contains only finitely many subformations (saturated subformations). A. N. Skiba [7] proved this assertion for each group G whose soluble residual G S has no Frattini G-chief factors. On the other hand, R. M. Bryant and P. D. Foy [8] proved that form G contains only finitely many subformations for each extension G of a soluble group by a simple nonabelian group. L. A. Shemetkov and A. N. Skiba [4] proved that an n-multiply saturated one-generated formation l n form G contains finitely many n-multiply saturated subformations provided that the soluble residual G S of G has no Frattini G-chief factors. At the same article, they proved that the one-generated totally saturated formation l ∞ form G, when G is soluble, contains finitely many totally saturated subformations. In the author’s article [9], it was proved that an analogous statement for a one-generated totally saturated formation l ∞ form G fails in general. Along with the above-mentioned questions of formation theory, the problem remains open of the equivalence of the finiteness condition for a subformation lattice of a multiply saturated formation and the length finiteness condition for this lattice (for example, see [10, Problem 9]). Extending the results of [9], we prove the next Theorem. Let F be a τ -closed totally saturated formation. Then the following are equivalent: (a) the lattice L τ ∞ (F) is of finite length; (b) the lattice L τ ∞ (F) is finite; (c) F is a soluble one-generated totally saturated formation. 2. Definitions and Notation All groups under study are finite. We use the terminology of [4,11]. Here, we recall some definitions and notation. Let A and B be groups, let ϕ : A → B be an epimorphism, and let Ω and Σ be some systems of subgroups in A and B, respectively. We denote by Ω ϕ the set {H ϕ | H ∈ Ω}, and we denote by Σ ϕ -1 the set {H ϕ -1 | H ∈ Σ} of all preimages in A of all groups in Σ. Gomel ′ . Translated from Sibirski˘ ı Matematicheski˘ ı Zhurnal, Vol. 48, No. 1, pp. 185–191, January–February, 2007. Original article submitted November 30, 2005. 150 0037-4466/07/4801–0150 c  2007 Springer Science+Business Media, Inc.