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Kang and Liu [‘On supersolvability of factorized finite groups’, Bull. Math. Sci.3 (2013), 205–210] investigate the structure of finite groups that are products of two supersoluble groups. The goal of this note is to give a correct proof of their main theorem.
By
A. Ballester-Bolinches, Departament d'Àlgebra, Universitat de Valéncia, C/ Doctor Moliner 50, 46100 Burjassot (Valéncia), Spain,
John Cossey, Mathematics Department, School of Mathematical Sciences, The Australian National University, Canberra, 0200, Australia,
M. C. Pedraza-Aguilera, ETS de Informática Aplicada, Departamento de Matemática Aplicada-IMPA, Universidad Politécnica de Valencia, Camino de Vera, s/n, 46022 Valencia, Spain
The study of factorized groups has played an important role in the theory of groups. We can consider so relevant results as the Ito's Theorem about products of abelian groups or the celebrated Theorem of Kegel-Wielandt about the solubility of a product of two nilpotent groups. In the very much special case when the factors are normal and nilpotent, a well-known result due to Fitting shows that the product is nilpotent. Nevertheless it is not true in general that the product of two normal supersoluble subgroups of a group is a supersoluble group. To create intermediate situations it is usual to consider products of groups whose factors satisfy certain relations of permutability. Following Carocca [12] we say that a group G = AB is the mutually permutable product of A and B if A permutes with every subgroup of B and vice versa. If, in addition, every subgroup of A permutes with every subgroup of B, we say that the group G is a totally permutable product of A and B.
In this context, we can consider as seminal the following results of Asaad and Shaalan.
Theorem A (Asaad and Shaalan [2]) (i) Assume that a group G = AB is the mutually permutable product of A and B. Suppose that A and B are supersoluble and that either A, B or G', the derived subgroup of G, is nilpotent. Then G is supersoluble.
By
A. Ballester-Bolinches, Departament d'Àlgebra, Universitat de València, C/ Doctor Moliner 50, 46100 Burjassot (València), Spain,
John Cossey, Mathematics Department, School of Mathematical Sciences, The Australian National University, Canberra, 0200, Australia,
Xiuyun Guo, Department of Mathematics, Shanxi University, Taiyuan, Shanxi 030006, People's Republic of China,
M. C. Pedraza-Aguilera, Departamento de Matemática Aplicada, E.U.I., Universidad Politécnica de Valencia, Camino de Vera, s/n, 46071 Valencia, Spain; The work of the first and last authors is supported by Proyecto PB97-0674-C02-02 of DGICYT, MEC, Spain
The well-known fact that a product of two normal supersoluble subgroups of a group is not necessarily supersoluble shows that the saturated formation of supersoluble groups need not be closed under the product of normal subgroups. This makes interesting the study of factorized groups whose subgroup factors are connected by certain permutability properties. Baer (see [2]) proved that if a group G is the product of two normal supersoluble subgroups, then G is supersoluble if and only if the commutator subgroup of G is nilpotent. This result has been generalized by Asaad and Shaalan ([1]) in the following sense: If G is the product of two subgroups H and K such that H permutes with every subgroup of K and K permutes with every subgroup of H, that is, G is the mutually permutable product of H and K, and G', the commutator subgroup of G is nilpotent, then G is supersoluble. Moreover they prove that in the case G = HK such that every subgroup of H permutes with every subgroup of K, that is, G is the totally permutable product of H and K, then if the factors H and K are supersoluble the group G is also supersoluble.
Further studies have been done by several authors within the framework of formation theory.
By
Manuel J. Alejandre, Centro de Investigación Operativa, Universidad Miguel Hernández, Avda. del Ferrocarril s/n, 03202 Elche, Spain,
A. Ballester-Bolinches, Departament d'Àlgebra, Universitat de València, C/ Doctor Moliner 50, 46100 Burjassot (València), Spain,
R. Esteban-Romero, Departamento de Matemática Aplicada, E.U.I., Universidad Politécnica de Valencia, Camino de Vera, s/n, 46071 Valencia, Spain,
M. C. Pedraza-Aguilera, Departamento de Matemática Aplicada, E.U.I., Universidad Politécnica de Valencia, Camino de Vera, s/n, 46071 Valencia, Spain; The work of the second and last authors is supported by Proyecto PB97-0674-C02-02 of DGICYT, MEC, Spain
A group G is said to be a T-group if every subnormal subgroup of G is normal in G, that is, if normality is a transitive relation. These groups have been widely studied (see [10], [11], or [14]).
A subgroup H of a group G is said to be permutable (or quasinormal) in G if HK = KH for all subgroups K of G. Permutability can be considered thus as a weak form of normality. The study of groups G in which permutability is transitive, that is, H permutable in K and K permutable in G always imply that H is permutable in G, has been a successful field of research in recent years. Such groups are called PT-groups. According to a theorem of Kegel [12, Satz 1], every permutable subgroup of G is subnormal in G. Consequently, PT-groups are exactly those groups in which subnormality and permutability coincide; that is, those groups in which every subnormal subgroup permutes with every other subgroup. Therefore, every T-group is clearly a PT-group.
One could wonder what would happen if we did not require that every subnormal subgroup of a group G permutes with any other subgroup of G, but only with a certain family of its subgroups. In this direction, those groups in which every subnormal subgroup of G permutes with every Sylow p-subgroup of G for each prime p have sometimes been called T*-groups (see [3]) or also (π – q)-groups (see [1]).
Only finite groups are considered here. The well-known fact that the product of two normal supersoluble subgroups of a group is not necessarily supersoluble shows that formations, even saturated, need not be closed under the product of normal subgroups. Clearly, any formation is, however, closed under direct products and even under central products. Therefore it is interesting to study factorized groups whose subgroup factors are connected by certain permutability properties. In fact, the following question can be formulated. Let the group G = HK be the product of subgroups H and K of G which lie in a formation ℱ. What is a relationship between the factors H and K -weaker than their elementwise permutability in the case of a direct product- which will guarantee G ∈ ℱ? In [1] the following results are proved:
Theorem 1Let G = HK be a group which is the product of two supersoluble subgroups H and K. If every subgroup of H permutes with every subgroup of K (we say in this case that H and K are totally permutable subgroups of G), then G is supersoluble.
Finite groups which are products of pairwise totally permutable subgroups are studied in this paper. The -residual, -projectors and -normalizers in such groups are obtained from the corresponding subgroups of the factor subgroups under suitable hypotheses.
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