A remark on a product of two formational tcc-subgroups

A subgroup 𝐴 of a group 𝐺 is called tcc-subgroup in 𝐺 , if there is a subgroup 𝑇 of 𝐺 such that 𝐺 = 𝐴𝑇 and for any 𝑋 (cid:54) 𝐴 and 𝑌 (cid:54) 𝑇 there exists an element 𝑢 ∈ ⟨ 𝑋, 𝑌 ⟩ such that 𝑋𝑌 𝑢 ≤ 𝐺 . The notation 𝐻 (cid:54) 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 , when 𝐴 and 𝐵 belong to F and F is a saturated formation such that U ⊆ F . Here U is the formation of all supersoluble groups.


Introduction
Throughout this paper, all groups are finite and always denotes a finite group. We use the standard notations and terminology of [1,2]. The notation means that is a subgroup of a group .
It is well known that the product of two normal nilpotent subgroups of a group is nilpotent. However, the product of two normal supersoluble subgroups of a group 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 supersolubility. A starting point of this research can be located at M. Asaad and A. Shaalan's paper [3]. In particular, they proved the supersolubility of a group = such that the subgroups and are totally permutable and supersoluble, see [3,Theorem 3.1]. Here the subgroups and of a group are totally permutable if every subgroup of is permutable with every subgroup of . 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 nonsaturated 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 4-5] 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 of a group is called tcc-subgroup in , if it satisfies the following conditions: 1) there is a subgroup of such that = ; 2) for any and there exists an element ∈ ⟨ , ⟩ such that ≤ . We say that the subgroup is a tcc-supplement to in . Now, we can state the main result in [10], which is the following: Theorem

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 and ̸ = , we write < . The notation ¢ means that is a normal subgroup of a group . Denote by ( ), ( ) and Φ( ) the centre, Fitting and Frattini subgroups of respectively, and by O ( ) the greatest normal -subgroup of . Denote by ( ) the set of all prime divisors of order of . The semidirect product of a normal subgroup and a subgroup is written as follows: . The monographs [11], [12] contain the necessary information of the theory of formations. A formation F is said to be saturated if /Φ( ) ∈ F implies ∈ F. In view of Theorems 3.
If is a subgroup of , then = ⋂︀ ∈ is called the core of in . If a group contains a maximal subgroup with trivial core, then is said to be primitive and is its primitivator. A simple check proves the following lemma. Lemma 1. Let F be a saturated formation and be a group. Assume that / ∈ F, but / ∈ F for all non-trivial normal subgroups of . Then is a primitive group.
Recall that the product = is said to be tcc-permutable [7], if for any and there exists an element ∈ ⟨ , ⟩ such that ≤ . The subgroups and in this product are called tcc-permutable. Lemma 2. ([7, Theorem 1, Proposition 1-2]) Let = be the tcc-permutable product of subgroups and and be a minimal normal subgroup of . Then the following statements hold: where is a prime.  (1) is a tcc-subgroup in for any subgroup of such that ; (2) / is a tcc-subgroup in / for any ¢ ; (3) for every 1 ¢ and ≤ there exists an element ∈ such that 1 ≤ . In particular, 1 ≤ for some maximal subgroup of and 1 ≤ for some Hall -subgroup of soluble and any ⊆ ( ); (4) 1 ≤ for every subnormal subgroup of and for every 1 ¢ ; (5) if ¢ such that ≤ and ∩ = 1, then 1 ¢ for every 1 ¢ such that 1 ≤ ; (6) if ¢ such that ∩ = 1 and ≤ , then 1 ≤ Let ̸ = 1 be a proper tcc-subgroup in a primitive group and be a tcc-supplement to in . Suppose that is a unique minimal normal subgroup of . If ∩ = 1 and ≤ , then is a cyclic group of order dividing − 1.
Proof. Since ∩ = 1 and ≤ , by Lemma 4 (6), ≤ ( ) for any ¢ . By Lemma 5, is an elementary abelian group. We fix an element ∈ . If ∈ , then ∈ ⟨ ⟩, since ≤ Therefore we can assume that = for all ∈ , where 1 ≤ ≤ and is a positive integer. Hence we have induces a power automorphism group on . By the Fundamental Homomorphism Theorem, / ( ) is isomorphic to a subgroup of P( ), where P( ) is the power automorphism group of . Since is abelian, it follows that ( ) = by [2, Theorem 4.41] and ( ) = 1. On the other hand, P( ) is a cyclic group of order − 1. Really P( ) is a group of scalar matrices over the field P consisting of elements. Hence P( ) is isomorphic to the multiplicative group P * of P and besides, P * is a cyclic group of order − 1. Therefore is a cyclic group of order dividing − 1. P Lemma 7. Let F be a formation, group, and subgroups of such that and belong to F.
be the external direct product of groups and . Since ∈ F, ∈ F and F is a formation, we have × ∈ F. Let : × → be a function with (( , )) = . It is clear that is a surjection. Because it follows that is an epimorphism. The core Ker contains all elements ( , ) such that = 1. In this case = −1 ∈ ∩ ( ). By the Fundamental Homomorphism Theorem, Since × ∈ F and F is a formation, × /Ker ∈ F. Hence ∈ F. P Lemma 8. ([14, Lemma 2.16]) Let F be a saturated formation containing U and be a group with a normal subgroup such that / ∈ F. If is cyclic, then ∈ F.

Proof of Theorem 2
Assume that the claim is false and let be a minimal counterexample. Suppose that is simple. By Lemma 3,and are normal in , a contradiction. Hence let be an arbitrary non-trivial normal subgroup of . The quotients / ≃ / ∩ and / ≃ / ∩ are tcc-subgroups in / by Lemma 4 (2), / ≃ / ∩ ∈ F and / ≃ / ∩ ∈ F, because F is a formation. Hence the quotient / = ( / )( / ) ∈ F by induction. Since F is a saturated formation, it follows that Φ( ) = 1, has a unique minimal normal subgroup and is primitive by Lemma 1. By Lemma 5, is abelian and ( ) = = ( ) = O ( ), = , where | | = and is a primitivator. By Lemma 2, is either | | = , or ≤ and ∩ = 1, or ∩ = 1 and ≤ , where is a tcc-supplement to in . In the first case, by Lemma 8, ∈ F. Suppose that ≤ and ∩ = 1. Since is a tcc-subgroup in , it follows that by Lemma 6, is a cyclic group of order dividing − 1. Then ∈ ( ), where is the canonical local definition of U. Since U ⊆ F, we have by [12,Proposition IV.3.11], ( ) ⊆ ( ), where is the canonical local definition of F. Hence ∈ ( ). Let be a Sylow -subgroup of . It is obvious that ≤ for some Sylow subgroup of . Then we can always choose a primitivator of such that ≤ . Really = and ≤ = for some ∈ and some Sylow -subgroup of . It is clear that is a maximal subgroup of . If ≤ , then = = = = , a contradiction. Hence = . Because is abelian, then ∩ = 1 and is a primitivator.
Hence we have the following result.