Influence of weakly H-subgroups of minimal subgroups on the structure of finite groups

Let G be a finite group. A subgroup H of G is called an H-subgroup in G if NG(H) ∩ H g ≤ H for all g ∈ G. A subgroup H of G is called a weakly H-subgroup in G if there exists a normal subgroup K of G such that G = HK and H ∩K is an H-subgroup in G. In this paper, we use weakly H-subgroup condition on minimal subgroups to study the structure of the finite group G. Some earlier results are improved and extend.


Introduction
Throughout this paper, all groups are finite. Our notation is standard and taken mainly from Doerk and Hawkes [8].
A question of particular interest in the theory of groups is to study the structure of a group G by using a certain generalized normality of some subgroups of G. Wang [16] introduced the concept of c-normality of a subgroup of a finite group as follows: A subgroup H of a group G is said to be c-normal in G if G has a normal subgroup K of G such that G = HK and H ∩ K ≤ H G , where H G = ∩ g∈G H g is the core of H in G, that is, the largest normal subgroup of G contained in H. Bianchi et al. [5] introduced the following concept: A subgroup H of a group G is said to be an H-subgroup in G if N G (H) ∩ H g ≤ H for all g ∈ G. In [2], Asaad, Heliel and Al-Shomrani introduced a new subgroup embedding property of a finite group, called a weakly H-subgroup, which is a generalization of both c-normality and H-subgroup. A subgroup H of a group G is said to be a weakly H-subgroup in G if there exists a normal subgroup K of G such that G = HK and H ∩K ∈ H(G), where H(G) denotes the set of all H-subgroups of a group G. It is clear that each Example 1.1. Set H = S 5 , the symmetric group of degree 5, and L = A 5 , the alternating group of degree 5. Let K be a group of order 2 and consider G = H ×K. Let P be a Sylow 2subgroup of H and P 1 be a Sylow 2-subgroup of L . Clearly, P is not c-normal in G as H = S 5 is a non solvable group. On the other hand, A minimal subgroup of a group G is a subgroup of prime order. How minimal subgroups can be embedded in a group G is a question of particular interest in studying the structure of G. In fact, many authors have investigated the influence of normality, c-normality, H-subgroup and more recently weakly H-subgroup of the minimal subgroups of a group G on the structure of G; see for example [1], [3,4], [6,7], [10], [12,20] and [22,23]. The present paper may be viewed as a continuation of [1], [17] and [20]. More precisely, we study the influence of weakly H-subgroups of the minimal subgroups of a group G on its structure. Some earlier results are improved and extend.

Basic definitions and preliminaries
In this section, for the sake of convenience, we collect some definitions and state some known results from the literature which will be used in the sequel.
Recall that a class of groups F is a formation provided that the following conditions are satisfied: (1) If G ∈ F, then G/N ∈ F, where N is any normal subgroup of G.
(2) If G/M and G/N are both in F, then G/(M ∩ N ) is also in F for any normal subgroups M and N of G.
A formation F is said to be saturated if G/Φ(G) ∈ F implies that G belongs to F. Throughout this paper, U and R will denote the classes of supersolvable groups and nilpotent groups, respectively. It is known that U and R are saturated formations.
A normal subgroup N of a group G is an F-hypercentral subgroup of G provided N possesses a chain of subgroups 1 = N 0 N 1 ... N s = N such that N i+1 /N i is an F-central chief factor of G (see [8], p. 387). The product of all F-hypercentral subgroups of G is again an F-hypercentral subgroup, denoted by Z F (G), and called the F-hypercenter of G (see [8], IV 6.8). For the formation R of nilpotent groups, the R-hypercenter of a group G is simply the terminal member Z ∞ (G) of the ascending central series of G.
Let G be a minimal non p-nilpotent group (A non p-nilpotent group all of whose proper subgroups are p-nilpotent), where p is a prime. Then (i) G is a minimal non-nilpotent group (A non-nilpotent group all of whose proper subgroups are nilpotent).
(ii) G = P Q, where P is a normal Sylow p-subgroup of G and Q is a non normal cyclic Sylow q-subgroup of G.
(iv) If p > 2, then the exponent of P is p and, when p = 2, the exponent of P is at most 4.

Results
We first prove the following result: Proof: Let G be a counterexample of minimal order. Then we have: (1) G is a minimal non p-nilpotent group.
Let H be any proper subgroup of G, and let L be a cyclic subgroup of H of order p (or of order Thus H satisfies the hypothesis of the theorem in any case. The minimal choice of G implies that H is p-nilpotent. Thus G is a minimal non p-nilpotent group. From Lemma 2.2, G is a minimal non-nilpotent group, G = P Q, P G, Q G and P/Φ(P ) is a minimal normal subgroup of G/Φ(P ).
(2) p = 2 and every element of order 4 is a weakly H-subgroup in G.
If not, p > 2. By Lemma 2.2(iv), the exponent of P is p. Thus, by hypothesis, P ≤ Z ∞ (G). By Lemma 2.3, O p (G) ≤ C G (P ) and so G = P Q = P × Q is nilpotent, a contradiction. If every element of order 4 of G lies in Z ∞ (G), then again P ≤ Z ∞ (G) and we have a contradiction.
By (1), < x > Q =< x > ×Q and hence < x >≤ N G (Q). Thus, P ≤ N G (Q), a final contradiction completing the proof of the theorem. ✷ Since a group G is nilpotent if and only if it is p-nilpotent for every prime p dividing the order of G, the following corollaries are immediate consequences of Theorem 3.1:   (2) P ≤ N . Clearly, from (1), G/P is nilpotent (in particular p-nilpotent) and, since G/N is p-nilpotent, we have that G/(P ∩N ) is p-nilpotent. Now if P N , then P ∩N < P and so Q(P ∩ N ) < QP = G. Thus, from (1), Q(P ∩ N ) is nilpotent and hence Q(P ∩ N ) = Q × (P ∩ N ). Since G/(P ∩ N ) = (P/(P ∩ N ))(Q(P ∩ N )/(P ∩ N )) is p-nilpotent, we have that Q(P ∩ N )/(P ∩ N ) G/(P ∩ N ) and so Q(P ∩ N ) G. Now Q is characteristic in Q(P ∩ N ) G implies that Q G, a contradiction. Thus P ≤ N .
(3) Final contradiction. If p > 2, then, by Lemma 2.2(iv), the exponent of P is p. Thus P = P ∩ N ≤ Z ∞ (G). By Lemma 2.3, O p (G) ≤ C G (P ) and so G = P Q = P ×Q, a contradiction. Assume that p = 2. Clearly, as P G, every element of order 2 or of order 4 of G is contained in P (in particular in N by (2)). Hence every element of order 2 of G lies in Z ∞ (G) and every cyclic subgroup of order 4 of G is a weakly H-subgroup in G or lies in Z ∞ (G) by hypothesis. Applying Theorem 3.1 yields G is 2-nilpotent, a contradiction completing the proof of the theorem. ✷ The following corollaries are immediate consequences of Theorem 3.5: Corollary 3.6. Let G be a group and N be a normal subgroup of G such that G/N is nilpotent. If every minimal subgroup of N is contained in Z ∞ (G) and every cyclic subgroup of order 4 of N is a weakly H-subgroup in G or lies in Z ∞ (G), then G is nilpotent.
Corollary 3.7. Let G be a group and N be a normal subgroup of G such that G/N is nilpotent. If every minimal subgroup of N is contained in Z ∞ (G) and every cyclic subgroup of order 4 of N is an H-subgroup in G or lies in Z ∞ (G), then G is nilpotent. Corollary 3.10. Let G be a group and assume that every cyclic subgroup of order subgroup in G " is replaced by "every subgroup of prime order of F (H) is a weakly H-subgroup in G". The following example illustrates that: Example 3.11. Let F = NU be the class of groups G whose commutator subgroup G´is nilpotent. Clearly, F is a formation. If G/Φ(G) ∈F, then (G/Φ(G))´= G´Φ(G)/Φ(G) is nilpotent and therefore G´Φ(G) is nilpotent. Then G´≤ G´Φ(G) implies that G´is nilpotent. Thus F is saturated. Also the class F contains the class U as the commutator subgroup of a supersolvable group is nilpotent. Now consider the group G = GL (2,3). This group has a normal subgroup H isomorphic to the quaternion group of order 8 such that G/H ∼ = S 3 , the symmetric group of order 6, and therefore we have G/H ∈F. Notice that the unique subgroup of F (H) with prime order is the center of H and this is not only a weakly H-subgroup in G but also normal in G. Since the commutator subgroup of G = GL(2, 3) is G´= SL(2, 3), then G / ∈F.
In view of example 3.11, we prove the following theorem: Proof . (i) Let G be a counterexample of minimal order. Since G is 2-nilpotent, it follows that H is also 2-nilpotent and therefore H = P K, where P is a Sylow 2-subgroup of H and K is a normal Hall 2´-subgroup of H. As K is characteristic in H and H G, we have K G. Clearly, (G/K)/(H/K) ∼ = G/H ∈ F, F (H/K) is 2group and G/K is 2-nilpotent. Thus G/K satisfies the hypothesis of the theorem. By minimality of G, G/K ∈ F. But F (K) ≤ F (H), then applying Lemma 2.5 yields G ∈ F, a contradiction completing the proof of (i).
(ii) Let G be a counterexample of minimal order and let P be an abelian Sylow 2-subgroup of G. For every element x ∈ Ω 1 (P ∩ F (H)), consider the minimal 2subgroup P * of G contained in L =< x g : g ∈ G > which is the minimal normal subgroup of G containing x. Then P * ≤ F (H). By hypothesis, P * is a weakly Hsubgroup in G. Then there exists a normal subgroup K of G such that G = P * K and P * ∩K ∈ H(G). Clearly, G = LK and L∩K = 1or L∩K = L as L is a minimal normal subgroup of G. If L ∩ K = 1, then L = L ∩ (P * K) = P * (L ∩ K) = P * . In other words, P * =< x g > for some g ∈ G. Hence x g lies in the center of G and so does x. This means that Ω 1 (P ∩ F (H)) lies in the center of G. By Lemma 2.6, every element of odd order centralizes P ∩ F (H) and so P ∩ F (H) lies in the center of G as P is abelian. Now, clearly, G/(P ∩ F (H)) satisfies the hypothesis of the theorem. The minimal choice of G implies that G/P ∩F (H) ∈ F. Applying Lemma 2.5 yields G ∈ F, a contradiction. Thus we may assume that L ∩ K = L. Then L ≤ K and so P * = P * ∩ K ∈ H(G). By Lemma 2.4, P * G and hence P * = L which yields G ∈ F by the above arguments, a final contradiction completing the proof of (ii).
The following corollaries are immediate consequences of Theorem 3.12: Corollary 3.13. [20], Theorem 4Let F be a saturated formation containing the class of supersolvable groups U and let G be a group with a normal solvable subgroup H such that G/H ∈ F. Then G ∈ F under either of the following:.  Corollary 3.14. Let F be a saturated formation containing the class of supersolvable groups U and let G be a group with a normal solvable subgroup H such that G/H ∈ F. Then G ∈ F under either of the following:.