Products of finite connected subgroups

For a non-empty class of groups $\cal L$, a finite group $G = AB$ is said to be an $\cal L$-connected product of the subgroups $A$ and $B$ if $\langle a, b\rangle \in \cal L$ for all $a \in A$ and $b \in B$. In a previous paper, we prove that for such a product, when $\cal L = \cal S$ is the class of finite soluble groups, then $[A,B]$ is soluble. This generalizes the theorem of Thompson which states the solubility of finite groups whose two-generated subgroups are soluble. In the present paper our result is applied to extend to finite groups previous research in the soluble universe. In particular, we characterize connected products for relevant classes of groups; among others the class of metanilpotent groups and the class of groups with nilpotent derived subgroup. Also we give local descriptions of relevant subgroups of finite groups.


Introduction and main results
All groups considered in this paper are assumed to be finite. We take further previous research on the influence of two-generated subgroups on the structure of groups, in interaction with the study of products of subgroups. In [13] the following result is proven: (2) For all primes p = q, all p-elements a ∈ A and all q-elements b ∈ B, a, b is soluble. Obviously, for the special case A = B = G, the following well-known result of J. Thompson is derived: [18,6]

) A finite group G is soluble if and only if every two-generated subgroup of G is soluble.
Thompson's theorem has been generalized and sharpened in various ways. We mention in particular the extension of R. Guralnick, K. Kunyavskiȋ, E. Plotkin and A. Shalev, which describes the elements in the soluble radical G S of a finite group G. Theorem 1.3. (Guralnick, Kunyavskiȋ, Plotkin, Shalev, [15]) Let G be a finite group and let x ∈ G. Then x ∈ G S if and only if the subgroup x, y is soluble for all y ∈ G.
Again the application of Theorem 1.1, with A = G, and B = x ≤ G, assures that x G S is a normal (soluble) subgroup of G under the hypothesis in statement (1), and implies Theorem 1.3.
This shows how an approach involving factorized groups provides a more general setting for local-global questions related to two-generated subgroups. A first extension of Thompson's theorem for products of groups was obtained by A. Carocca [4], who proved the solubility of S-connected products of soluble subgroups. This way the following general connection property turns out to be useful: [3]) Let L be a non-empty class of groups. Subgroups A and B of a group G are L-connected if a, b ∈ L for all a ∈ A and b ∈ B. If G = AB we say that G is the L-connected product of the subgroups A and B.
Structure and properties of N -connected products, for the class N of finite nilpotent groups, are well known (cf. [1,14,2]); for instance, G = AB is an N -connected product of A and B if and only if G modulo its hypercenter is a direct product of the images of A and B. Apart from the above-mentioned results regarding S-connection, corresponding studies for the classes N 2 and N A of metanilpotent groups, and groups with nilpotent derived subgroup, respectively, have been carried out in [8,9]; in [10] connected products for the class S π S ρ of finite soluble groups that are extensions of a normal π-subgroup by a ρ-subgroup, for arbitrary sets of primes π and ρ, are studied. The class S π S ρ appears in that reference as the relevant case of a large family of formations, named nilpotent-like Fitting formations, which comprise a variety of classes of groups, such as the class of π-closed soluble groups, or groups with Sylow towers with respect to total orderings of the primes. A study in [11] of connected subgroups, for the class of finite nilpotent groups of class at most 2, contributes generalizations of classical results on 2-Engel groups.
In the present paper, as an application of Theorem 1.1, we show that main results in [8,9,10], proved for soluble groups, remain valid for arbitrary finite groups. In particular, we characterize connected products for some relevant classes of groups (see Theorem 1.6). For instance, we prove that for a finite group G = AB, the subgroups A and B are N 2 -connected if and only if A/F (G) and B/F (G) are N -connected, which means that for all a ∈ A and b ∈ B, a, b N ≤ F ( a, b ) if and only if for all a ∈ A and b ∈ B, a, b N ≤ F (G), where for any group X, F (X) denotes the Fitting subgroup of X, and X N denotes the nilpotent residual of X, i.e. the smallest normal subgroup of X with nilpotent quotient group. When we specialize our results to suitable factorizations, as mentioned above, we derive descriptions of the elements in F k (G), the radical of a group G for the class N k of soluble groups with nilpotent length at most k ≥ 1, as well as the elements in the hypercenter of G modulo F k−1 (G), in the spirit of the characterization of the soluble radical in Theorem 1.3 (see Corollaries 1.13, 1.11). In particular, this first result contributes an answer to a problem posed by F. Grunewald, B. Kunyavskiȋ and E. Plotkin in [16]. These authors present a version of Theorem 1.3 for general classes of groups with good hereditary properties [16,Theorem 5.12], by means of the following concepts: For a class X of groups and a group G, an element g ∈ G is called locally Xradical if g x belongs to X for every x ∈ G; and the element g ∈ G is called globally X -radical if g G belongs to X .
For a subset S and a subgroup X of a group G, we set S X = s x | s ∈ S, x ∈ X , the smallest X-invariant subgroup of G containing S. For g ∈ G, we write g X for {g} X . When X is a Fitting class, the property g x ∈ X is equivalent to g ∈ g, x X , the X -radical of g, x , as the property g G ∈ X is equivalent to g ∈ G X , and these properties are useful in the problem of characterizing elements forming G X . As mentioned in [16, Section 5.1], a main problem is to determine classes X for which locally and globally Xradical elements coincide. Corollary 1. 13 gives a positive answer for the class N k of finite soluble groups of nilpotent length at most k ≥ 1.
When the Fitting class X is in addition closed under extensions and contains all cyclic groups, the condition g x ∈ X is equivalent to g, x ∈ X , but this is not the case for important classes of groups, as the class N of finite nilpotent groups, or more generally N k , k ≥ 1. In this situation the condition g, x ∈ X for all elements x ∈ G may well not be equivalent to g ∈ G X , but still of interest as shown in Corollary 1.11 in relation with the hypercenter.
We shall adhere to the notation used in [5] and we refer also to that book for the basic results on classes of groups. In particular, π(G) denotes the set of all primes dividing the order of the group G. Also A and S π , π a set of primes, denote the classes of abelian groups and soluble π-groups, respectively. For the class of all finite π-groups, the residual of any group X is denoted O π (X), and O π (X) stands for the corresponding radical of X. If F is a class of groups, then N F is the class of groups which are extensions of a nilpotent normal subgroup by a group in F.
We gather next our main results. The first one extends to the universe of finite groups results for soluble groups in  3. Let π, ρ be arbitrary sets of primes. The following are equivalent: Remark 1.7. In Theorem 1.6 (3), (b) and (c), As consequences of Theorem 1.6 we derive Corollaries 1.8, 1.9, 1.11, 1.13, and point out again that corresponding results for finite soluble groups were firstly obtained in [8,Corollaries 1,2,3,4].  Then G ∈ N F implies A, B ∈ N F.
As a particular case of Corollary 1.8 we state explicitly: Corollary 1.9. If the group G = AB is the N 2 -connected product of the π-separable subgroups A and B of π-length at most l, π a set of primes, then G is π-separable of π-length at most l + 1.
Remark 1. 10. Easy examples show that the bound for the π-length of G in Corollary 1.9 is sharp. For instance, for any l ≥ 1, consider a set of primes π = ∅ with π ′ = ∅, where π ′ stands for the complement of π in the set of all prime numbers, let B be a π-separable group of π-length l such that O π (B) = 1, let p ∈ π and A be a faithful module for B over the field of p elements. Let G = [A]B be the corresponding semidirect product of A with B. Then A and B are N 2 -connected and the π-length of G is l + 1.
Corollary 1.11. Let G be a group, g ∈ G and k ≥ 1. Then g, h ∈ N k for all h ∈ G if and only if g ∈ Z ∞ (G mod F k−1 (G)).
1. For k = 1, Corollary 1.11 gives a characterization of the hypercenter of a group. This particular case appears already in [8, Corollary 3] as a direct consequence of Lemma 2.1 (2) below, and was also observed by R. Maier, as mentioned in [16,Remark 5.5], and referred to [17].
2. Assume that g ∈ G such that g, x is soluble for all x ∈ G. Let l be the highest nilpotent length of all these subgroups, so that g, x ∈ N l for all x ∈ G. By Corollary 1.11 it follows that g ∈ F l (G) ≤ G S . So that Corollary 1.11 may be seen also as generalization of the characterization of the soluble radical in Theorem 1.3.
Corollary 1.13. For a finite group G, an element g ∈ G and k ≥ 1, the following statements are equivalent: 3. g ∈ F k (G), i.e. g is globally N k -radical. 2. For k = 1, the Baer-Suzuki theorem states that F (G) = {g ∈ G | g, g x ∈ N for every x ∈ G}. But for k = 2, one can not conclude that g ∈ F 2 (G) whenever g, g x ∈ N 2 for all x ∈ G, as pointed out by Flavell [7]. Remark 1.15. As application of Theorem 1.6, the hypothesis of solubility can be also omitted in Corollary 4 and Propositions 3, 4 of [9], especially in relation with saturated formations F ⊆ N A, such as the class of supersoluble groups. Also an extension for finite groups of Corollary 1 of [10], in relation with the above-mentioned nilpotent-like Fitting formations, can be stated.

2.
A ∩ B ≤ Z ∞ (G), the hypercenter of G. F is either a saturated formation or a formation containing N , and A, B ∈ F, then G ∈ F. Proof. Since [A, B] ≤ G S and B a G S = G, it follows that a G S G. Therefore a ≤ G S and BG S = G. For any x ∈ A, we have now that B x G S = G, and so again x ∈ G S , which implies A ≤ G S . Definition 2.4. We define a subset functor T to assign to each finite group G a subset T (G) of G satisfying the following conditions:  Proof. Let G = AB be a finite group, A, B ≤ G, such that [A, B] ≤ G S , and assume that P (A, B, G) holds. Let a ∈ S 1 (A), b ∈ S 2 (B). We aim to prove that a, b Y ≤ G X . We argue by induction on |G|. Assume first that G S a B = G. Then A ≤ G S by Lemma 2.3. If G = G S b , then G is soluble and the result follows. So we may assume that

If
The proposition is proved. Remark 2.7. As we will see, Proposition 2.6 provides the main tool to derive Theorem 1.6 from Theorem 1.1 and the corresponding previous results in the soluble universe. In Notation 2.5 (2), the additional restriction of subgroups L = (L∩A)(L∩B) to subgroups of the form L = (L∩A)(L∩B) = G S X y , {X, Y } = {A, B}, y ∈ Y , will be required only for the application to the proof of Part (4) of Theorem 1.6, as it is also the case of the following Lemma 2.8. The present formulations of Notation 2.5 and Proposition 2.6 unify the treatment of the different parts stated in Theorem 1.6. Lemma 2.8. A is a normally embedded subgroup of A is a normally embedded subgroup of a group G = AB Proof. 1. Let p ∈ π(A). We consider a = a p a p ′ , where a p , a p ′ denote the p-part and the p ′ -part of a, respectively. Let M p ∈ Syl p (A) such that a p ∈ M p . By the hypothesis, there exists M ✂ G such that M p ∈ Syl p (M ). We claim that (N ∩M p ) a p ∈ Syl p (N a ∩A). Since N ∩A A, we have that N ∩M p ∈ Syl p (N ∩A). Consequently, (N ∩M p ) a p ∈ Syl p ((N ∩A) a p ). Since N a ∩ A = (N ∩ A) a = (N ∩ A) a p a p ′ , the claim follows easily.

Assume that
We prove next that (N ∩ M p ) a p ∈ Syl p (N a ∩ M ). Since N a ∩ M N a , it will follow that N a ∩ A is normally embedded in N a , which will conclude the proof.
We notice that N a ∩ M = (N a p ′ ∩ M ) a p , so that it is enough to prove that N ∩ M p ∈ Syl p (N a p ′ ∩ M ).
Again On the other hand, N p ∈ Syl p (N a p ′ ) and N a p ′ ∩ M N a p ′ , which implies that N p ∩ M ∈ Syl p (N a p ′ ∩ M ), and we are done.
2. Since [A, B] ≤ G S , we have that BG S G = AB = BG S A. The result follows now from part 1.
3. Set N = BG S G = AB, as before. Notice that