On factorizations of limited solubly ω-saturated formations

On factorizations of limited solubly ω-saturated formations

If F = F₁…Ft is the product of the formations F₁,…,Ft and F ≠ F₁…Fi−₁Fi+₁…Ft for all i = 1,…,t, then we call this product a non-cancellative factorization of the formation F. In this paper we gives a description of factorizable limited solubly ω-saturated formations.

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Дата:2012
Автор: Selkin, V.
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Опубліковано: Інститут прикладної математики і механіки НАН України 2012
Назва видання:Algebra and Discrete Mathematics
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Цитувати:On factorizations of limited solubly ω-saturated formations / V. Selkin // Algebra and Discrete Mathematics. — 2012. — Vol. 13, № 2. — С. 289–298. — Бібліогр.: 11 назв. — англ.

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spelling irk-123456789-1522112019-06-09T01:25:19Z On factorizations of limited solubly ω-saturated formations Selkin, V. If F = F₁…Ft is the product of the formations F₁,…,Ft and F ≠ F₁…Fi−₁Fi+₁…Ft for all i = 1,…,t, then we call this product a non-cancellative factorization of the formation F. In this paper we gives a description of factorizable limited solubly ω-saturated formations. 2012 Article On factorizations of limited solubly ω-saturated formations / V. Selkin // Algebra and Discrete Mathematics. — 2012. — Vol. 13, № 2. — С. 289–298. — Бібліогр.: 11 назв. — англ. 1726-3255 2010 MSC:20D10. http://dspace.nbuv.gov.ua/handle/123456789/152211 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
language English
description If F = F₁…Ft is the product of the formations F₁,…,Ft and F ≠ F₁…Fi−₁Fi+₁…Ft for all i = 1,…,t, then we call this product a non-cancellative factorization of the formation F. In this paper we gives a description of factorizable limited solubly ω-saturated formations.
format Article
author Selkin, V.
spellingShingle Selkin, V.
On factorizations of limited solubly ω-saturated formations
Algebra and Discrete Mathematics
author_facet Selkin, V.
author_sort Selkin, V.
title On factorizations of limited solubly ω-saturated formations
title_short On factorizations of limited solubly ω-saturated formations
title_full On factorizations of limited solubly ω-saturated formations
title_fullStr On factorizations of limited solubly ω-saturated formations
title_full_unstemmed On factorizations of limited solubly ω-saturated formations
title_sort on factorizations of limited solubly ω-saturated formations
publisher Інститут прикладної математики і механіки НАН України
publishDate 2012
url http://dspace.nbuv.gov.ua/handle/123456789/152211
citation_txt On factorizations of limited solubly ω-saturated formations / V. Selkin // Algebra and Discrete Mathematics. — 2012. — Vol. 13, № 2. — С. 289–298. — Бібліогр.: 11 назв. — англ.
series Algebra and Discrete Mathematics
work_keys_str_mv AT selkinv onfactorizationsoflimitedsolublyōsaturatedformations
first_indexed 2025-07-13T02:33:19Z
last_indexed 2025-07-13T02:33:19Z
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fulltext Jo ur na l A lg eb ra D is cr et e M at h.Algebra and Discrete Mathematics RESEARCH ARTICLE Volume 13 (2012). Number 2. pp. 289 – 298 c© Journal “Algebra and Discrete Mathematics” On factorizations of limited solubly ω-saturated formations Vadim M. Selkin Communicated by L. A. Shemetkov Abstract. If F = F1 . . .Ft is the product of the formations F1, . . . ,Ft and F 6= F1 . . .Fi−1Fi+1 . . .Ft for all i = 1, . . . , t, then we call this product a non-cancellative factorization of the formation F. In this paper we gives a description of factorizable limited solubly ω-saturated formations. Introduction All groups considered are finite. We will use Cp(G) to denote the intersection of all centralizers of Abelian p-chief factors of the group G [1] (we note that Cp(G) = G if G has no such chief factors). Let X be a set of groups. Then we use Com(X) to denote the class of all Abelian groups A such that A ≃ H/K for some composition factor H/K of some group G ∈ X. Also, we write Com(G) for the set Com({G}). Let ∅ 6= ω ⊆ P. For every function f of the form f : ω ∪ {ω′} → {group formations} (1) we put CFω(f) = {G is a group | G/R(G) ∩ Oω(G) ∈ f(ω′) 2010 MSC: 20D10. Key words and phrases: factorizations, solubly ω-saturated formation, compo- sition ω-satelitte, one-generated formation. Jo ur na l A lg eb ra D is cr et e M at h. 290 On factorizations of limited solubly formations and G/Cp(G) ∈ f(p) for any prime p ∈ ω ∩ π(Com(G)}. Here R(G) denotes the radical of G (i.e. R(G) is the largest normal soluble subgroup of G). We call F a solubly ω-saturated formation [2] if F = CFω(f) for some function f of the form (1). In this case, we call f a composition ω-satelitte of the formation F. If F = F1 . . .Ft (2) is the product of the formations F1, . . . ,Ft and F 6= F1 . . .Fi−1Fi+1 . . .Ft for all i = 1, . . . , t, then we call (2) a non-cancellative factorization of the formation F. A formation F is called a one-generated formation if there is a group G such that F is the intersection of all formations containing G. A formation F is called a one-generated solubly ω-saturated formation if there is a group G such that F is the intersection of all solubly ω-saturated formations containing G. A formation F is called limited if F is a subformation of some one- genereted formation. Analogously, a solubly ω-saturated formation F is called a limited solubly ω-saturated formation if it is a subformation of some one-generation solubly ω-saturated formation. Let H be a class of groups. We use H(ω′) to denote the class form(A/(R(A) ∩ Oω(R)) | A ∈ H). and use H(p) to denote the class form(A/Op(A) | A ∈ H). In this paper we prove the following theorem which gives the answer to Problem 21 in [2]. Theorem 1. The product F1F2 . . .Ft (∗) is a non-cancellative factorization of some limited solubly ω-saturated formation F if and only if the following conditions hold: (1) t ≤ 3 and every factor in (*) is a nonidentity formation; (2) F1 is a one-generated ω-saturated subformation in NωN and π(Com(F)) ∩ ω ⊆ π(F1); (3) If F1 * Nω, then t = 2, F2 is an Abelian one-generated formation and for all groups A ∈ F1 and B ∈ F2, (|A/Fω(A)|, |B|) = 1 and (|A/Oω(A)|, |B|) = 1; Jo ur na l A lg eb ra D is cr et e M at h. V. Selkin 291 (4) If F1 ⊆ Nω and t = 3, then |π(F1| > 1, F3 is a one-generated Abelian formation and for every p ∈ π(F1), the formation F2(p) is one-generated nilpotent and for all groups A ∈ F2 and B ∈ F3, π(A/Op(A)) ∩ π(B) = ∅; (5) If F1 ⊆ Nω, t = 2 and |π(F1)| > 1, then F2(ω′), F2 are limited formations; (6) If F1 = Np for some prime p, then F2(ω′) and F2(p) (if p ∈ ω) are limited formations, F2 * F1, and there is a group B ∈ F2 such that for all groups A ∈ F1, the F2-residual T F2 of the wreath product T = A ≀ B is contained subdirectly in the base group of T . All unexplained natation and terminology are standard. The reader is referred to [3], [4], [5] if necessary. 1. Proof of Theorem Proof. Suppose that F = F1F2 . . .Ft and the conditions (1)–(6) hold. Our first step is to show that F is a limited solubly ω-saturated formation. In view of Lemma 5 [2] and Theorem 1 [6], we only need to show that f(a) is a limited formation for all a ∈ π(Com(F))∩ω∪{ω′}, where f is the smallest composition ω-satelitte of the formation F, and |π(Com(F)) ∩ ω| < ∞. By (2), we see that |π(Com(F)) ∩ ω| < ∞. Let H = F2 . . .Ft, and m be the smallest composition ω-satelitte of the formation F1. Then by Lemma 4.5 [7], we have F = CFω(t), where t(a) =        m(p)H, if a = p ∈ π(Com(F1)) ∩ ω ∅, if a = p ∈ ω \ π(Com(F1)) m(ω′)H, if a = ω′. Let F1 * Nω. Then by hypothesis, H = F2 is a one-generated Abelian formation. Let p ∈ π(Com(F1)) ∩ ω. Since F1 is a one-generated solubly ω-saturated formation and F1 ⊆ NωN, then m(p) is a nilpotent one- generated formation. Since for any groups A ∈ F1 and B ∈ F2 we have (|A/Fp(A)|, |B|) = 1, then by Lemma 5 [2], m(p) ∩ H = (1). Then m(p)H is soluble formations and it is is a one-generated formation. But f(p) ⊆ t(p) = m(p)H. Hence f(p) is a limited formation. Analogously, we can show that f(ω′) is a limited formation. Hence, F is indeed a limited solubly ω-saturated formation. Jo ur na l A lg eb ra D is cr et e M at h. 292 On factorizations of limited solubly formations Let F1 ⊆ Nω. In this case, by Theorem 2 [6], we only need to show that if t = 3, then H(ω′) and H(p) (for all p ∈ π(F1)) are limited formations and H is a one-generated formation if |π(F)| > 1. Let p ∈ π(F1) ∩ ω. Consider the formation F2(p)F3. In view of Con- dition (4) we have π(F2(p)) ∩ π(F3) = ∅. Besides F2(p) is a nilpotent one-generated formation. In these cases, the product F2(p)F3 is a one- generated formation. But evidently F2(p)F3 is a soluble formation, and every subformation of F2(p)F3 is also one-generated. Thus, in order to prove that (F2F3)(p) is a limited formation, we only need to show that it is a subformation in F2(p)F3. Let A ∈ F2F3. Then AF3 ∈ F2. Hence Op(AF3) is a characteristic subgroup of AF3 such that AF3/Op(AF3) ∈ F2(p). But AF3/Op(AF3) = (A/Op(AF3))F3 , we have A/Op(AF3) ∈ F2(p)F3, and so A/Op(A) ∈ F2(p)F3. Thus (F2F3)(p) ⊆ F2(p)F3. This shows that the formation (F2F3)(p) is limited. Assume that |π(F)| > 1 and let p, q ∈ π(F1). Let A ∈ F2 and B ∈ F3. Since |A/Op(A)|, |B|) = 1 and (|A/Oq(A)|, |B|) = 1, we have (|A|, |B|) = 1. This shows that the exponents of the formations F2 and F3 are coprime. Same as above, one can show that F2 is a nilpotent formation. Hence by Theorem 1 [8], F2F3 is a limited formation. Consider the formation F2(ω′)F3. Clearly, F2(ω′) is a one-generated nilpotent formation and π(F2(ω′)) ∩ π(F3) = ∅. Hence F2(ω′)F3 is a soluble one-generated formation. Now, in order to prove that (F2F3)(ω′) is a limited formation, we only need to show that (F2F3)(ω′) ⊆ F2(ω′)F3. Let A ∈ F2F3. Since Oω(AF3) is a characteristic subgroup of AF3 such that AF3/(Oω(AF3) ∈ F2(ω′), and so A/Oω(A) ∈ F2(ω′)F3. Hence (F2F3)(ω′) ⊆ F2(ω′)F3. We still need to show that the factorization (*) is non-cancellative. For this purpose, we first take t = 2. Assume that F1 * Nω. Then by Conditions (3), F2 is an Abelian formation, and so by Lemma 5.1 [7], F 6= F2. Suppose that F = F1. And let A be a group with minimal order in F1 \Nω. Let R be the monolith of A. Then R = ANω . Clearly, we have R * Φ(A). Let B be a simple group in F2 and T = A ≀ B = [K]B where K is the base group of T . Since F2 is Abelian, we have B = Cp. Assume that the F2-residual T F2 of the wreath product T = A ≀ B is not contained subdirectly in the base group of T . Let π(T F2) is a projection of T F2 in A1, where A1 is the first copy of A in K. Then N/π(T F2) ≀B is a homomorphic image of the group T/T F2 . By (3), |N/Oω(N), |A|) = 1. This contradiction Jo ur na l A lg eb ra D is cr et e M at h. V. Selkin 293 shows that the F2-residual T F2 of the wreath product T = A≀B is contained subdirectly in the base group of T . Hence T F2 ≃ A ∈ F1. So T ∈ F = F1 ⊆ NωN. It is clear that R = F (A) and by Lemma 3.1.9 [5], we deduce that L = R♮ = ∏ b∈B Rb 1 = F (T ) is the monolith of T , where R1 is the monolith of the first copy A in K. Since T ∈ NωN, we have TN ∈ Nω, i.e. TN ⊆ L. Let R be an ω′- group. Hence L is an ω′-group, and so Oω(T ) = 1. Since T ∈ NωN, T must be a nilpotent group. But F (T ) = L 6= T , which is a contradiction. Hence R is a p-group, for some p ∈ ω. However, since A /∈ Nω, we have R = Fω(A) and consequently, (|A/R|, |B|) = 1. Let B be a q-group. Then B is a Sylow q-subgroup of T1 = (A/R) ≀ B. By [[1], A, (18.2)], we have T1 ≃ T/L. This proves that T1 is a nipotent group. Thus, B E T , and so B ∩ K1 6= 1, where K1 is the base group of T1, a contradiction. This shows that F1 6= F 6= F2. Now we assume that F1 ⊆ Nω. Let |π(F1)| > 1 and p, q be different primes in π(F1). Also we let B be a group such that F2 ⊆ formB. Since F1 is an ω-local formation, we have N{p,q} ⊆ F1. Hence F 6= F2 by Lemma 3.1.5 [5]. In view of Lemma 5.1 [7], we conclude that F 6= F1. Let π(F1) = {p} for some p ∈ ω. Then F1 = Np. Let B be a group in F2 such that for every group A ∈ F1 the F2-residual T F2 of the wreath product T = A ≀B is contained subdirectly in the base group of T . Assume that F = F2 = NpF2 and let A be a non-identity group in Np. If T = A ≀B, then T ∈ F = F2, and so T F2 = 1 is not contained subdirectly in the base group of T . This contradiction shows that F 6= F2. And since, by Condition (6), F2 * F1, then we have F 6= F1. This shows that the factorization (*) is indeed non-cancellative. Now let t = 3. Consider the case π(F1) = {p}. By (6) F2 6⊆ Np. Let A be a group of minimal order in F2 \Np. Then Op(A) = 1. Thus, if B ∈ F3, we have (|A|, |B|) = 1. Let T = Zp ≀ (A ≀ B). Evidently, T ∈ F and T is not a metanilpotent group. Hence F 6⊆ N2. Since the formations F1F2, F1F3 are all metanilpotent, F 6= F1F2,F1F3. By (6) we can let B be a group in F2 such that for all non-identity groups A ∈ F1 the F2-residual DF2 of the group D = A ≀ B is contained subdirectly in the base group of the wreath product D. Let C be a non-identity group in F3. Assume that F2F3 = F1F2F3 and T = D ≀ C = [K]C, where K is the base group Jo ur na l A lg eb ra D is cr et e M at h. 294 On factorizations of limited solubly formations of D. Then since F3 is Abelian formation, we see that T F3 is contained subdirectly in K, and so D ∈ F2, that is, DF2 = 1, a contradiction. Hence F 6= F2F3. Now assume |π(F1)| > 1. Let {p, q} ⊆ π(F1). By (4), F2(p) and F2(q) are a one-generated nilpotent formations. Hence, F2 is also a one-generated nilpotent formation by Lemma 4.6 [7]. Therefore F 6= F2F3 by Lemma 5.1 [7]. For the case |π(F1)| > 1, the proof is similar. Assume that F = F1F2 . . .Ft ⊆ cωform(G) = F∗ for some group G. Let f be the smallest composition ω-satelitte of the formation F, f∗ be the smallest composition ω-satelitte of the formation F∗. We will show that the factors of the non-cancellative factorization (*) all satisfy Conditions (1)–(6). It is clear that every factor in (*) is a non-identity formation. In additions, by Lemma 5.3 [7], we have t ≤ 3. Thus, (1) is true. By Theorem 1 [9] we have F1 ⊆ NωN, hence F1 is a hereditary formation. Then F1 ⊆ F ⊆ F∗. By Lemma 3.5 [7] and Proposition A [7] we see that F1 is an ω-saturated soluble formation. But by Lemma 4 [10] the set of all ω-saturated subformations of the formation F1 is finite. Hence, there is a chain (1) = M0 ⊆ M1 ⊆ . . . ⊆ Mn−1 ⊆ Mn = F1, where Mi is a maximal ω-saturated subformation of Mi+1, i = 0, 1, . . . , n− 1. Let Ai ∈ Mi\Mi−1, i = 1, 2, . . . , n. Assume that lωform(Mi−1∪{Ai}) 6= Mi. Then Mi−1 ⊆ lωform(Mi−1 ∪ {Ai}) ⊆ Mi. But Mi−1 is a maximal ω-saturated subformation of Mi. This contradic- tion shows that lωform(Mi−1 ∪ {Ai}) = Mi, and so F1 = lωform(A1, . . . , An) = lωform(A1 × . . . × An) is a one-generated ω-saturated formation. Assume that p is prime such that p ∈ π(Com(F)) ∩ ω and p /∈ π(F1). Then Np ⊆ F and Zp /∈ F1. Let Zq be a simple group in F1 such that q ∈ π(Com(F1)) ∩ ω . Then q 6= p. Let B be a cycle group of order pm, where m = |G|. And let D = Zq ≀ B = [K]B, where K is the base of the wraeth product D. In is clear that D ∈ F, and hence D/Cq(D) = D/K ≃ B ∈ f(q). Jo ur na l A lg eb ra D is cr et e M at h. V. Selkin 295 By Lemma 5 [2] we see f(q) ⊆ f∗(q) = form(G/Cq(G)), which is a contradition to Lemma 3.1.5 [5]. Hence π(Com(F))∩ω ⊆ π(F1). This shows that (2) holds. Assume that F1 * Nω. Let H = F2 . . .Ft and A be a group of minimal order in F1\Nω. Let R be the monolith of A. Then R = ANω = CA(R) and R * Φ(A). If R is a p-group, then R = Op(A) = Fp(A). By (2), A/R is a nilpotent group. But by Lemma 1.7.11 [11], we have Op(A/CA(R)) = 1, and so p /∈ π(A/R). Assume that R = A. Then, we have |R| = p and R = ANω . We can deduce that p /∈ ω. Hence H is an Abelian formation by Lemma 3.1 [7]. Let R 6= A and let R ≤ M ≤ A, where M is a maximal subgroup of A. Then A/M is a simple group with |A/M | 6= p. Let m be the smallest local ω-satelitte of F1. Since A ∈ F1, we have A/Fp(A) = A/R ∈ m(p), and so A/M ∈ m(p). Now, by using Lemma 5.2 [7], we see that t = 2 and that F2 is an Abelian formation. It follows that F = F1F2 is a solubly ω-saturated soluble formation and hence, in this case, F is an ω-saturated formation. Now we prove that F2 is a one-generated formation. Assume that π(F1) ∩ ω = {p}. Since F1 6⊆ Nω, we may choose in F1 a group A of prime order q 6∈ ω. Let B ∈ F2 and T = A ≀ B. Then, it is clear that T ∈ F and that Oω(T ) = Op(T ) = 1. Hence by Lemma 5 [2], we have T ≃ T/Oω(T ) ∈ f(ω′) ⊆ f∗(ω′) = form(G/Oω(G)). Hence F2 ⊆ from(G/Oω(G)). Since the formation F2 is soluble, F2 is a one-generated formation. Now let |π(F1) ∩ ω| > 1 and let p, q be two different primes in π(F1) ∩ ω. Let B ∈ F2. Then by Lemma 4.5 [7], we have F = CF (t), where t(a) =        m(p)F2 if a = p ∈ π(Com(F1)) ∩ ω; ∅ if a = p ∈ ω \ π(Com(F1)); m(ω′)F2 if a = ω′. We note that the formation function m is an inner composition ω-satelitte of F1. Now, using Lemma 5 [2], we deduce that B/Op(B) ∈ f(p) ⊆ f∗(p) = = form(G/Cp(G)) ⊆ form(G/Cp(G) × (G/Cq(G)) Jo ur na l A lg eb ra D is cr et e M at h. 296 On factorizations of limited solubly formations and B/Oq(B) ∈ f(q) ⊆ f∗(q) = = form(G/Cq(G)) ⊆ form(G/Cp(G) × (G/Cq(G)). Hence B ≃ B/(Op(B) ∩ Oq(B)) ∈ form((G/Cp(G)) × (G/Cq(G))). This shows that F2 ⊆ form(G/Cp(G) × (G/Cq(G)) and so F2 is a one- generated formation. Let A ∈ F1 and B ∈ F2. Then by Lemma 3.1.5 [5] and Lemma 3.1.7 [5] we can show that (|A/Fω(A)|, |B|) = 1 and (|A/Oω(A)|, |B|) = 1. This proves the condition (3). Next we assume that F1 ⊆ Nω and t = 3. First consider the case where |π(F1)| > 1. We claim that F2(p) is a nilpotent formation, for all p ∈ ω. In fact, if the claim is not true, then we can let A be a non-nilpotent group of smallest order in F2(p). Let R be the monolith of A. And let q be a prime in π(F1) such that R * Oq(A). Let T = Zq ≀ A, where Zq is a group of order q. Then T ∈ F1(F2(p)) ⊆ F1F2 ⊆ NωN. Also, it is not difficult to show that F (T ) = K, where K is the base group of T . Clearly, T/K ≃ A is not nilpotent, and so T /∈ NωN. This contradiction shows that F2(p) is a nilpotent formation, and our claim is established. If F1F2 ⊆ Nω, then by Lemma 5.1 [7], we have F1F2 = F1 = Np which is impossible. Hence F1F2 * Nω and so by (3), F3 is an Abelian one- generated formation. Let p ∈ π(F1), A ∈ F2, T0 = A/Op(A) and B ∈ F3. Assume that there is prime q such that q ∈ π(T0)∩π(B). Let Zq be a group of order q. Let T = Zq ≀ (Z1 × . . . × Zn), where Z1 ≃ Z2 ≃ . . . ≃ Zn ≃ Zq and n = |G|. Form Y = T0 ≀ (B1 × . . . × Bn), where Bi ≃ B. Then, by Lemma 3.1.7 [5], the nilpotent class c(T ) of the group T is at least n + 1. If q /∈ ω, then by [[1], A, (18,2)] and by Lemma 5 [2], T/Oω(T ) ≃ T ≃ E ≤ Y/Oω(Y ) ∈ form(G/Oω(G)). This clearly contradicts to Lemma 3.1.5 [5]. Let q ∈ ω. Then, in view of (2), we have q ∈ π(F1) \ {p}. Let Zp be a group of order p and D = Zp ≀ T = [K]T , where K is the base group of D. Clearly D ∈ F1F2F3 = F, and so by Lemma 5 [2], we have D/Cp(D) = D/K ≃ T ∈ form(G/Cp(G)), Jo ur na l A lg eb ra D is cr et e M at h. V. Selkin 297 again, this contradicts to Lemma 3.1.5 [5]. Thus, for all groups A ∈ F2 and B ∈ F3, we have π(A/Op(A)) ∩ π(B) = ∅. Now we claim that F2(p) is a one-generated nilpotent formation. Indeed, by (2), F1F2 is a one-generated ω-saturated formation. By using Proposition 4.7 [7] we see that F2 or F2(p) is a one-generated formation. But F2 is a soluble formation and F2(p) ⊆ F2. Hence F2(p) is a one- generated formation in view of Theorem VII.1.7 [1]. Now consider F1 = Np, for some p ∈ ω. By Proposition 4.7 [7], F2(p) is a one-generated formation since F1F2 is a one-generated solubly ω- saturated formation. Thus, condition (4) holds. Condition (5) and the first two conditions of (6) follow directly from Proposition 4.7 [7]. It is clear that F2 * F1. Now we assume that for every group B ∈ F2 there is a group A ∈ F1 such that the F2-residual T F2 of the wreath product T = A ≀ B is not contained subdirectly in the base group of T . And let B be a group of minimal order in NpF2 \F2. Then the group B is monolithic and its monolith R = BF2 . Now let T = A ≀ (B/R), where A is a group in F1 such that the F2-residual T F2 of T is not contained subdirectly in the base group K of T . But then the formation F2 contains the group Zp ≀ (B/R), where |Zp| = p. Evidently, R is an elementary Abelian p-group, so by Lemma 3.5.2 [5], B ∈ F2. This contradiction shows that NpF2 ⊆ F2. Hence NpF2 = F2 = F. This contradiction shows that condition (6) holds. Thus, the theorem is proved. References [1] K. Doerk and T. Hawkes, Finite Soluble Group, Walter de gruyter, Berlin–New York, 1992. [2] A.N.Skiba and L.A.Shemetkov, Multiply L-composition formations of finite groups, Ukrainsk. Math. Zh., 52(6), 2000, 783–797. [3] L.A.Shemetkov, Formations of Finite Groups, Nauka, Moscow, 1978. [4] L.A.Shemetkov, A.N.Skiba, Formations of Algebraic Systems, Nauka, Moscow, 1989. [5] A.N.Skiba, Algebra of Formations, Minsk, Belaruskaja Navuka, 1997. [6] V.M.Selkin, Criterions of a limitation of solubly ω-saturaed formations of finite groups, Vesti NAN Belarusi, 2012, 2, p.131-136. [7] W.Go, V.M.Selkin, K.P.Sham Factorization theory of one-generated Bear ω-local formations, Communications in Algebra, Volume 35, Issue 9 September 2007, p. 2901-2931 [8] A.N.Skiba, On product of formations. Algebra and Logic, 22(5), 1983, 574–583. [9] V.M.Selkin, About one application of the theory of minimal τ -closed ω-saturated non H-formations, Vesnik VSU, v3, 2012, 46-52. Jo ur na l A lg eb ra D is cr et e M at h. 298 On factorizations of limited solubly formations [10] V.M.Selkin, On one criterion of product of nonidentity formations, Proc. GGU, v2, 2012, p.23-27. [11] Wenbin Guo, The Theory of Classes of Groups, Science Press-Kluwer Academic Publishers, Beijing-New York-Dordrecht-Boston-London, 2000. Contact information V. Selkin Department of Mathematics, Francisk Skorina Gomel State University, Sovetskya Str., 104, Gomel, 246019, Belarus E-Mail: vselkin@gsu.by Received by the editors: 25.02.2012 and in final form 25.02.2012. V. Selkin

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