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ЧЕБЫШЕВСКИЙ СБОРНИК
Том 22. Выпуск 1.
УДК 512.542 DOI 10.22405/2226-8383-2021-22-1-495-501
Замечание о произведении двух формационных tcc-подгрупп1
А. А. Трофимук
Александр Александрович Трофимук — кандидат физико-математических наук, Брестский государственный университет им. A.C. Пушкина (Беларусь, г. Брест). e-mail: alexander. trofimuk@gmail. com
Аннотация
Подгруппа А группы G называется tcc-подгрушюй в G, если существует подгруппа Т группы G такая, что G = AT и для любого X ^ А и Y ^ Т существует элемент u € (X,Y) такой, что XYU < G. Садись Н < G означает, что Н является подгруппой группы G. В этой статье мы исследуем группу G = AB при условии, что А и В являются tcc-подгруппами в G. Доказано, что такая группа G припадлежит F, если подгруппы А и В принадлежат F где F — насыщенная формация такая, что U С F Здесь U — формация всех сверхразрешимых групп.
Ключевые слова: сверхразрешимая группа, тотально перестановочное произведение, насыщенная формация, tcc-перестановочное произведение, tcc-подгруппа.
Библиография: 15 названий. Для цитирования:
А. А. Трофимук. Замечание о произведении двух формационных tcc-подгрупп // Чебы-шевский сборник, 2021, т. 22, вып. 1, с. 495-501.
CHEBYSHEVSKII SBORNIK Vol. 22. No. 1.
UDC 512.542 DOI 10.22405/2226-8383-2021-22-1-495-501
A remark on a product of two formational tcc-subgroups
A. A. Trofimuk
Alexander Alexandrovich Trofimuk — candidate of physical and mathematical sciences, Brest State A.S. Pushkin University (Belarus, Brest). e-mail: alexander. trofimuk®gmail. com
1 Исследование выполнено при финансовой поддержке Белорусского республиканского фонда фундаментальных исследований (проект Ф19РМ-071).
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A. A. TpckJmimvk
Abstract
A subgroup A of a group G is called tec-subgroup in G, if there is a subgroup T of G such that G = AT and for any X < A and Y < T there exists an element u e (X, Y) such that XYU < G. The notation H < G means that H is a subgroup of a group G. In this paper we consider a group G = AB such that A and B are tcc-subgroups in G. We prove that G belongs to F 11 A and B belong to F and F is a saturated formation such that U C F Here U is the formation of all supersoluble groups.
Keywords: supersoluble group, totally permutable product, saturated formation, tcc-permutable product, tcc-subgroup.
Bibliography: 15 titles. For citation:
A. A. Trofimuk, 2021, "A remark on a product of two tcc-subgroups" , Chebyshevskii sbornik, vol. 22, no. 1, pp. 495-501.
1. Introduction
Throughout this paper, all groups are finite and G always denotes a finite group. We use the standard notations and terminology of fl, 2]. The notation H ^ G means that H is a subgroup of a group G.
It is well known that the product of two normal nilpotent subgroups of a group G is nilpotent. However, the product of two normal supersoluble subgroups of a group G is not necessarily supersoluble. It seems then natural to consider factorized groups in which certain subgroups of the corresponding factors permute, in order to obtain new criteria of supersolubilitv. A starting point of this research can be located at M. Asaad and A. Shaalan's paper [3]. In particular, they proved the supersolubilitv of a group G = AB such that the subgroups A and B are totally permutable and supersoluble, see [3, Theorem 3.1]. Here the subgroups A and B of a group G are totally permutable if every subgroup of A is permutable with every subgroup of B. In [4] Maier showed that this statement is also true for the saturated formations containing the formation U of all supersoluble groups. Ballester-Bolinches and Perez-Ramos in [5] extend Maier's result to non-saturated formations which contain all supersoluble groups. This direction have since been subject of an in-depth study of many authors, see, for example, [6], [7], [8]. The monograph [9, chapters 45] contains other detailed information on the structure of groups, which are totally or mutually permutable products of two subgroups.
The following concept was introduced in [8] .
Definition . A subgroup A of a group G is called tcc-subgroup in G, if it satisfies the following conditions:
1) there is a subgroup T of G such that G = AT;
2) for any X ^ A and Y ^ T there exists an element u e (X, Y) such that XYu < G.
We say that the subgroup T is a tcc-supplement to A in G.
Now, we can state the main result in [10], which is the following:
Th eorem 1. ([10, Theorem A]) Let, G = AB, where A and B are tcc-subgroups in G. Let F be a saturated formation of soluble groups such that U C F Suppose that A and B belong to F Then G belongs to F
In this article we show that the hypothesis of solubility in Theorem 1 can be removed.
Th eorem 2. Let G = AB, where A and B are tcc-su,bgroups in G. LetF be a saturated form,ation such that U C F Suppose that A and B belong to F Then G belongs to F
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. If H ^ G and H = G, we write H < G. The notation H < G means that H is a normal subgroup of a group G. Denote by Z(G), F(G) and the centre, Fitting and Frattini subgroups of G respectively, and by
Op(G) the greatest normal p-subgroup of G. Denote by k(G) the set of all prime divisors of order of G. The semidirect product of a normal subgroup A and a subgroup B is written as follows: A x B.
The monographs [11], [12] contain the necessary information of the theory of formations. A formation F is said to be saturated if G/§(G) £ F implies G £ F- In view of Theorems 3.2 and 4.6 in [12, IV], for any non-empty saturated formation F there exists a formation function f (that is, any function of the form f : P ^ {formations}) such that F = LF(f) := [G | G/FP(G) £ f (p) for all primes p dividing |G|}. Here FP(G) = Op',p(G) is the greatest normal p-nilpotent subgroup of G [12, IV, Section 7]. Such a function is called a local definition of F- Moreover, in view of
F
f (called the canonical local definition of F) such th at f (p) = Npf (p) C F fa all primes p, where Npf (p) = 0 if f (p) = 0 and Npf (p) is the class of all groups A with Af (*) < Op(A) whenever f (P) = 0
If H is a subgroup of G, then Hq = HxeG is called the core of H in G. If a group G contains a maximal subgroup M with trivial core, then G is said ^o he primitive and M is its primitivator. A simple check proves the following lemma.
Lemma 1. Let F be a saturated formation and G be a group. Assume that G £ F, bu,t G/N £ F for all non-trivial normal subgroups N of G. Then G is a primitive group.
Recall that the product G = AB is said ^o be tcc-permutable [7], if for any X ^ A and Y ^ B there exists an element u £ (X, Y) such th at XYu < G. The subgroups A and B in this product are called tcc-permutable.
Lemma 2. ([7, Theorem 1, Proposition 1-2]) Let G = AB be the tcc-permutable product of subgroups A and B and N be a minimal normal subgroup of G. Then the following statements hold:
(1) [A n N,B n N} C [1,N};
(2) if N < A n B or N n A = N n B = 1, then IN| = p, where p is a prime.
Lemma 3. ([13, Theorem 4]) Let, G = AB be the tcc-permutable product of subgroups A and B. Then [A,B] < F'(G).
Lemma 4. ([8, Lemma 3.1]) Let A be a tcc-subgroup in G and Y be a tcc-supplement to A in G. Then the following statements hold:
(1) A is a tcc-subgroup in H for any subgroup H of G such that A ^ H;
(2) AN/N is a tcc-subgroup in G/N for any N < G;
(3) for every A\ < A and X < Y there exists an element y £ Y such that A\Xy < G. In particular, A\M < G for some maximal subgroup M of Y and A\H < G for some Hall 'K-subgroup H of soluble Y and any n C n(G);
(4) A\K < G for every subnormal subgroup K of Y and for every A\ < A;
(5) if T < G such that T < AandT n Y = 1, then Ti < G for every Ti < A such that Ti < T;
(6) if T < G such that T n A = 1 an d T < Y, then A1 < Nq(T1) for eve ry T1 < T and for every A1 < A.
Lemma 5. Let G be a group and N a unique minimal normal subgroup of G. If G has a proper tcc-subgroup A such that A = 1, then N is abelian.
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Proof. Since A is a tcc-subgroup, it follows that G = AY, A and Y are tcc-permutable. If [A, Y] = 1, then A < Cg(Y). It is clear A and Y are normal in G. Thus N < A n Y. By Lemma 2, IN I = p and N is abelian. Therefore [A, Y] = 1 and N < [A, Y] < F (G) = 1 by Lemma 3. Hence N is abelian. □
Lemma 6. Let A = 1 be a proper tcc-subgroup in a primitive group G and Y be a tcc-supplement to A in G. Suppose that N is a unique minimal normal subgroup of G. If N n A = 1 and N < Y, then A is a cyclic group of order dividing p — 1.
Proof. Since N n A = ^d N < ^^^^taa 4 (6), A < NG(K) for any K < N. By Lemma 5, N is an elementary abelian group. We fix an element a e A. \i x e N, then xa e (x), since A < Ng((x)) by hypothesis. Hence xa = xm'x, where mx is a positive integer and 1 < mx < p. If y e N \ {x}, then
(xy)a = (xy)mxy = xmxyymxy, (xy)a = xaya = xmxymy,
xmxyymxy = xmxymy, xmxy-mx = ymy-mxy = 1, mXy = mx = my.
Therefore we can assume that xa = xna for all x e N, where 1 < na < p md na is a positive integer. Hence we have A induces a power automorphism group on N. By the Fundamental Homomorphism Theorem, A/Ga(N) is isomorphic to a subgroup of P(N), where P(N) is the power automorphism group of N. Since N is abelian, it follows that Gg(N) = N by [2, Theorem 4.41] and Ga(N) = 1. On the other hand, P(N) is a cyclic group of order p — 1. Really P(N) is a group of scalar matrices over the field P consisting of p elements. Hence P(N) is isomorphic to the multiplicative group P* of P and besides, P* is a cyclic group of order p — 1. Therefore A is a cyclic group of order dividing p — 1. □
Lemma 7. Let F be a formation, G group, A and B subgroups of G such that A and B belong to F If [A, B] = 1, then AB e F
Proof. Since
[A,B] = ([a,b] I a e A, b e B) = 1, it follows that ab = ba fa all a e A, b e B. Let
A x B = {(a,b) I a e A, b e B},
(ai,bi)(a2, b2) = (a\a2, b\b2), Va\,a2 e A, b\,b2 e B —
be the external direct product of groups A and B. Since A e F B e F and F is a formation, we have A x B e F Let <p : A x B ^ AB be a function with <^((a, b)) = ab. It is clear that <p is a surjection. Because
tp((ai,bi)(a,2,b2)) = tp((aia,2,bi 62)) = 01026162 = = aibia2b2 = ip((ai,bi))ip((a2, 62),
it follows that (f is an epimorphism. The core Ker tp contains all elements (a, b) such that ab = 1. In this case a = e A n B ^ Z(G). By the Fundamental Homomorphism Theorem,
A x B/Ker (p = AB.
Since A x B e F and F is a formation, A x B/Ker p e F Hence AB e F □
Lemma 8. ([14, Lemma 2.16]) Let, F be a saturated formation containing U and G be a group with a normal subgroup E such that G/E e F If E is cyclic, then G e F-
3. Proof of Theorem 2
Assume that the claim is false and let G be a minimal counterexample. Suppose that G is simple. By Lemma 3, A and B are normal in G, a contradiction. Hence let K be an arbitrary non-trivial normal subgroup of G. The quotients AK/K ~ A/A n K and BK/K ~ B/B n K are tcc-subgroups in G/N by Lemma 4(2), AK/K ~ A/A n K £ F and BK/K ~ B/B n K £ F, because F is a formation. Hence the quotient G/K = (AK/K)(BK/K) £ F by induction.
Since F is a saturated formation, it follows that $(G) = 1 G has a unique minimal normal subgroup N and G is primitive by Lemma 1. By Lemma 5, N is abelian and F (G) = N = Cg (N) = Op(G), G = N x M, where ^| = pn and M is a primitivator.
By Lemma 2, is either ^| = p, or N < A and N n Y = 1, or N n A = 1 and N < Y, where
Y is a tcc-supplement to A in G. In the first case, by Lemma 8, G £ F- Suppose that N < A and N n Y = 1. Since Y is a tcc-subgroup in G, it follows that by Lemma 6, Y is a cyclic group of order dividing p — 1. Then Y £ g(p),where g is the canonical local definition of U. Since U C F, we have by [12, Proposition IV.3.11], g(p) C f (p), where f is the canonical local definition of F- Hence
Y £ f (p).
Let Q be a Sylow g-subgroup of Y. It is obvious that Q < Gq fa some Sylow sub group Gq of G. Then we can always choose a primitivator H of G such that Q < H. Really Gq = Mq and Gq < M9 = H for some g £ G and some Svlow ^-subgroup Mqoi M. ft is clear that H is a maximal subgroup of G.HN < ^^en G = NM = NM9 = NH = H, a contradiction. Hence NH = G. Because N is abelian, then N n H = 1 rnd H is a primitivator. Since A = A n G = A n NH = N (A n H), we have
G = AY = N (A n H )Y.
Prove that (A n H)Y is a primitivator of G. Since
[A n H,Q] < [A, Y] = F(G) = N
by Lemma 3 and [A n H,Q] < H, it follows that [A n H,Q] < H n N = 1. Therefore A n H < Cg(Q)= T.Bvsides Y < T. Then
T = T n G = T n N (A n H )Y = (A n H)Y (N n T).
It is obvious that N n T is normal in T and hence N n T is normal in G = N (A n H )Y = NT, since N is abelian. Thus is either N < T, or N n T = 1. In the first case, T = G and Q < Z(G), a contradiction. Otherwise, T = (A n H)Y and G = N x T. №nce T = (A n H)Y is a primitivator of G. Thus we can always choose a primitivator M1 of G such that G = N x M^ Y < M1 and Mi = (A n Mi)Y.
Because A £ F, it follows that A/FP(A) £ f (p). Since N = CG(N^d N < A, we have that N < FP(A) = F(A). Let Ni is a minimal normal subgroup of A such that Ni < N. Then F(A) < Ca(N1) by [2, Lemma 4.21]. Since A is a tcc-subgroup in G, it follows that by Lemma 4 (5), Ni is normal in G. Hence N = Ni and GA(Ni) = CA(N) = N. Then FP(A) = N and A n Mi ~ A/N £ f (p).
Since f (p) is a famation, A n Mi £ f (p), Y £ f (p) md [A n Mi,Y] = 1, it follows that Mi £ f (p) by Lemma 7. Because N £ Np, we have G £ Npf (p) = f (p) C F-
So, we assume that N n A = 1 and N < Y. Similarly, we can show that N n B = 1 and N < X, where X is a tcc-supplement to B in G. By Lemma 6, A and B are cyclic. Hence G is supersoluble and therefore G £ F- The theorem is proved.
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4. Conclusion
Clear that by condition 2 of Definition 1,G = AT is the tcc-permutable product of the subgroups A and T. If G = AB is the tcc-permutable product of subgroups A and Б, then the subgroups A and В are tcc-subgroups in G. The converse is false.
Example 1. The dihedral group G =< a > x < c>, lal = 12, Icl = 2 ([15], IdGroup=[24,6]) is the product of tcc-subgroups A =< a3c > of order 2 and В = < a10 > x < с > of order 12. But A and В are not tcc-permutable. Indeed, there are the subgroups X = A and Y =< с > of A and В respectively such that doesn't exist и e {X, Y) =< a3 > x < с > such that XYu < G.
Hence we have the following result.
Corollary 1. 1. Let A and В be tcc-subgroups in G and G = AB. If A and В are supersoluble, then G is supersoluble, ([8, Theorem 4.1])
2. Let F be a saturated formation containing H. Let the group G = HK be the tcc-permutable product of subgroups H and K. If H e F and К e F, then G e F ([13, Theorem 5]).
3. Suppose that A and В are supersoluble subgroups of G and G = AB. Suppose further that A and В are totally permutable. Then G is supersoluble, ([3, Theorem 3.1]).
СПИСОК ЦИТИРОВАННОЙ ЛИТЕРАТУРЫ
1. Huppert В. Endliche Gruppen I. Berlin-Heidelberg-New York: Springer, 1967.
2. Monakhov V. S. Introduction to the Theory of Finite Groups and Their Classes [in Russian]. Minsk: Vvsh. Shkola, 2006.
3. Asaad M., Shaalan A. On the supersolubilitv of finite groups // Arch. Math. 1989. Vol. 53. R 318-326.
4. Maier R. A completeness property of certain formations // Bull. Lond. Math. Soc. 1992. Vol. 24. P. 540-544.
5. Ballester-Bolinches A., Perez-Ramos M. D. A question of R. Maier concerning formations //J. Algebra. 1996. Vol. 182. P. 738-747.
6. Guo W., Shum K.P., Skiba A.N. Criterions of supersolubilitv for products of supersoluble groups
Publ. Math. Debrecen. 2006. Vol. 68, №3-4. P. 433-449.
7. Arroyo-Jorda M., Arroyo-Jorda P. Conditional permutabilitv of subgroups and certain classes of groups // Journal of Algebra. 2017. Vol. 476. P. 395-414."
8. Trofimuk A. A. On the supersolubilitv of a group with some tcc-subgroups // Journal of Algebra and Its Applications. 2021. 2150020 (18 pages).
9. Ballester-Bolinches A., Esteban-Romero R., Asaad M. Products of finite groups. Berlin: Walter de Gruvter, 2010.
10. Trofimuk A. A. Trofimuk A. A. On a product of two formational tcc-subgroups // Algebra and Discrete Mathematics. 2020. Vol. 30, № 2. P. 282-289.
11. Ballester-Bolinches A., Ezquerro L.M. Classes of Finite Groups. Dordrecht: Springer, 2006.
12. Doerk K., Hawkes T. Finite soluble groups. Berlin-New York: Walter de Gruvter, 1992.
13. Arrovo-Jorda М., Arrovo-Jorda P., Martinez-Pastor A., Perez-Ramos M.D. On conditional permutabilitv and factorized groups // Annali di Matematica Рига ed Applicata. 2014. Vol. 193. P.1123-1138.
14. Skiba A.N. On weakly s-permutable subgroups of finite groups //J. Algebra. 2007. Vol. 315. P.192-209.
15. Groups, Algorithms, and Programming (GAP), Version 4.11.0. [Электронный ресурс] // URL: http://www.gap-system.org (дата обращения 22.09.2020).
REFERENCES
1. Huppert, В. 1967, Endliche Gruppen I, Springer, Berlin-Heidelberg-New York.
2. Monakhov, V. S. 2006, Introduction to the Theory of Final Groups and Their Classes [in Russian], Vvsh. Shkola, Minsk.
3. Asaad, M. к Shaalan, A. 1989, "On the supersolubilitv of finite groups", Arch. Math., vol. 53, pp. 318-326.
4. Maier, R. 1992, "A completeness property of certain formations", Bull. bond. Math. Soc., vol. 24, pp. 540-544.
5. Ballester-Bolinches, A. к Perez-Ramos, M.D. 1996, "A question of R. Maier concerning formations", J. Algebra, vol. 182, pp. 738-747.
6. Guo, W., Shum, K.P. к Skiba, A.N. 2006, "Criterions of supersolubilitv for products of supersoluble groups", Publ. Math. Debrecen, vol. 68, no. 3-4, pp. 433-449.
7. Arrovo-Jorda, M. к Arrovo-Jorda, P. 2017, "Conditional permutabilitv of subgroups and certain classes of groups", Journal of Algebra, vol. 476, pp. 395-414.
8. Trofimuk, A.A. 2021, "On the supersolubilitv of a group with some tcc-subgroups", Journal of Algebra and Its Applications, 2150020 (18 pages).
9. Ballester-Bolinches, A., Esteban-Romero, R. к Asaad, M. 2010, Products of finite groups, Walter de Gruvter, Berlin.
10. Trofimuk, A. A. 2020, "On a product of two formational tcc-subgroups", Algebra and Discrete Mathematics, vol. 30, no. 2, pp. 282-289.
11. Ballester-Bolinches, А. к Ezquerro, L.M. 2006, Classes of Finite Groups, Springer, Dordrecht.
12. Doerk, К. к Hawkes, T. 1992, Finite soluble groups, Walter de Gruvter, Berlin-New York.
13. Arrovo-Jorda, M., Arrovo-Jorda, P., Martinez-Pastor, А. к Perez-Ramos, M.D. 2014, "On conditional permutabilitv and factorized groups", Annali di Matematica Рига ed Applicata, vol. 193, pp. 1123-1138.
14. Skiba, A.N. 2007, "On weakly s-permutable subgroups of finite groups", J. Algebra, vol. 315, pp. 192-209.
15. Groups, Algorithms, and Programming (GAP), Version 4.11.0. (2020). Available at http:// www.gap-system.org (accessed 22 September 2020).
Получено 22.09.2020 г.
Принято в печать 21.02.2021 г.
