Some New Applications of Weakly H-Embedded Subgroups of Finite Groups

A subgroup H of a finite group G is said to be weaklyH-embedded in G if there exists a normal subgroup T of G such that HG = HT and H ∩ T ∈ H(G), where HG is the normal closure of H in G, and H(G) is the set of all H-subgroups of G. In the recent research, Asaad, Ramadan and Heliel gave new characterization of p-nilpotent: Let p be the smallest prime dividing |G|, and P a non-cyclic Sylow p-subgroup of G. Then G is p-nilpotent if and only if there exists a p-power d with 1 < d < |P| such that all subgroups of P of order d and pd are weakly H-embedded in G. As new applications of weaklyH-embedded subgroups, in this paper, (1) we generalize this result for general prime p and get a new criterion for p-supersolubility; (2) adding the condition “NG(P) is p-nilpotent”, here NG(P) = {g ∈ G|Pg = P} is the normalizer of P in G, we obtain p-nilpotence for general prime p. Moreover, our tool is the weaklyH-embedded subgroup. However, instead of the normality of HG = HT, we just need HT is S-quasinormal in G, which means that HT permutes with every Sylow subgroup of G.


Introduction
Throughout this paper, "G is a group" always means that "G is a finite group". For convenience, one can refer to [1][2][3][4] for the definitions and notions in the paper.
The T-groups are defined as the groups G in which normality is a transitive relation, that is, if H K G, then H G. In 2000, Bianchi Gillio Berta Mauri, Herzog and Verardi [5] proved a characterization of soluble T-groups by means of H-subgroup: a subgroup H of a group G is called an H-subgroup in G if N G (H) ∩ H g ≤ H, for every element g ∈ G, where N G (H) = {x ∈ G|H x = H} is the normalizer of H in G. They proved that a group G is a supersolvable T-group if and only if every subgroup of G is an H-subgroup of G. Later, except for the exploration of T-groups, H-subgroups were widely used to character finite groups. Csörgö and Herzog [6] obtained that a group G is supersolvable if every cyclic subgroup of G of prime order or order 4 is an H-subgroup. Asaad [7] proved that a group G is supersolvable if every maximal subgroup of every Sylow subgroup of G is an H-subgroup. The set of all H-subgroups of a group G is denoted by H(G). Moreover, Guo and Wei [8] gave new characterization of p-nilpotent or supersolvable by assuming some subgroups of G of the same order all belong to H(G), which provide a unified version of the results mentioned above if the order of G is odd. Moreover, Li, Zhao and Xu [9] considered the case when G is of even order.
Recently, Asaad et al. [10] introduced a new subgroup embedding property called weakly H-subgroup, which generalizes both c-normality and H-subgroup, called weakly H-subgroup. Soon after, Asaad and Ramadan [11] gave the definition of weakly H-embedded subgroup. Please note that a subgroup H of G is said to be a weakly H-embedded subgroup (weakly H-subgroup) of G if there exists a normal subgroup T of G such that H G = HT (G = HT) and H ∩ T ∈ H(G), where H G is the normal closure of H in G. Clearly, c-normal subgroups, H-subgroups and weakly H-subgroups imply weakly H-embedded subgroups. However, the converse does not hold in general, see [11] (Examples 1.3, 1.4 and 1.5).
In fact, these subgroups were widely used to investigate the structure of finite groups. As a result, many interesting results have been subsequently obtained, such as [7,[10][11][12][13].
In the recent research about H-subgroups, Asaad, Ramadan, and Heliel gave a new characterization of p-nilpotency. Theorem 1. ( [12] Theorem A) Let p be the smallest prime dividing |G|, and P a non-cyclic Sylow p-subgroup of G. Then G is p-nilpotent if and only if there exists a p-power d with 1 < d < |P| such that all subgroups of P of order d and pd are weakly H-embedded in G.
However, according to this result, some natural questions arise: (1) If delete the condition "p is the smallest prime dividing |G|", can we claim that G is p-supersoluble? (2) Does there exist another condition to obtain p-nilpotence rather than "p is the smallest prime dividing |G|"? (3) As we know, the condition that HT is the smallest normal subgroup of G containing H, is too strict. Can we replace it by a weaker embedding subgroup property?
In this paper, we further explore weakly H-embedded subgroups and pay attention to Problem 1. However, instead of the normality of HT, we just consider HT is S-quasinormal in G. As we know, a subgroup K is S-quasinormal in G, means that K permutes with every Sylow subgroup P of G, that is KP = PK. However, for convenience, we also called it a weakly H-embedded subgroup, that is: A subgroup H of a group G is said to be weakly H-embedded in G if there exists a normal subgroup T of G such that HT is S-quasinormal in G and H ∩ T ∈ H(G).
As an application of these subgroups, we give a positive answer to Problem 1 in the class of p-soluble groups, for detail: Theorem 2. Let E be a p-soluble normal subgroup of a group G such that G/E is p-supersoluble, where p is a prime divisor of |E|. Let P be a Sylow p-subgroup of E. Suppose that P has a subgroup D with 1 |D| < |P| such that all subgroups of P of order |D| and p|D| are weakly H-embedded in G. When |D| = 1 and P is a non-abelian 2-group, we further assume that all cyclic subgroups of P of order 4 are weakly H-embedded in G. Then G is p-supersoluble.
Moreover, to avoid the condition "p is the smallest prime dividing |G|" of Theorem 1, we further prove that the conclusion holds if this condition is replaced by "N G (P) is p-nilpotent". Consequently, we give an answer to Problem 1.

Theorem 3.
Let E be a normal subgroup of G such that G/E is p-nilpotent, and P be a non-cyclic Sylow p-subgroup of E, where p is a prime dividing |E|. Assume that N G (P) is p-nilpotent and P has a subgroup D with order 1 < |D| < |P| such that all subgroups of P of order |D| and order p|D| are weakly H-embedded in G. Then G is p-nilpotent.
In the second section, we list some lemmas which will be useful for the proofs of the above results. The proofs of Theorems 2 and 3 are put in the third section. Some previously known results are generalized by our theorems, and we list some in the fourth section. (3) Assume that H is a p-group and N a normal p -subgroup of G. Then HN/N is weakly H-embedded in G/N.

Preliminaries
Proof. By the hypothesis, G has a normal subgroup T such that HT is S-quasinormal in G and H ∩ T ∈ H(G).
Recall that a class of groups F is called a formation if for every group G, every homomorphic The intersection of all formations containing the set {G/O p ,p (G)|G ∈ F} is denoted by F(p), and F(p) denotes the class of all groups G such that G F(p) is a p-group. Associated with a saturated formation F, there is a function f of the form f : P → {group formations}, where f (p) = F(p) for any prime p, which divides |G| for some G ∈ F, and f (p) = ∅ otherwise. The function f is called the canonical local satellite of F. For more detail, please turn to ([3] P. 3) or ( [2] Chap. IV, Theorem 3.7 and Definitions 3.9). Now we recall the subgroup Z F (G) of G, which is called the F-hypercenter of G. In fact, Z F (G) the product of all such normal subgroups N of G whose G-chief factors H/K satisfying (H/K) (G/C G (H/K)) ∈ F. Lemma 4. Let F be a saturated formation and f the canonical local satellite of F. Let P be a normal p-subgroup of G. Then P ≤ Z F (G) if and only if one of the following holds:

Proofs of Main Results
The following proposition plays an important role in the proof of Theorem 2.

Proposition 1.
Let P be a normal p-subgroup of a group G. Assume that P has a subgroup D satisfying 1 |D| < |P|, such that all subgroups of P of order |D| and p|D| are weakly H-embedded in G. When |D| = 1 and P is a non-abelian 2-group, we further assume that all cyclic subgroups of P of order 4 are weakly H-embedded in G. Then P ≤ Z U (G).
Proof. Assume by contradiction that (G, P) is a counterexample of minimal order |G| + |P|. We proceed via the following steps.
(1) P is not a minimal normal subgroup of G. Assume that P is minimal normal in G. Let H be a subgroup of P of order |D| or p|D|, which is normal in some Sylow subgroup of G. By the hypothesis, H is weakly H-embedded in G. So G has a normal subgroup T such that HT is S-quasinormal in G and H ∩ T ∈ H(G). Please note that P ∩ T is normal in G, so P ∩ T = 1 or P ∩ T = P by the minimality of P. If P ∩ T = 1, then H = H(P ∩ T) = P ∩ HT is S-quasinormal in G. However, by the choice of H and Lemma 1(2), H G, a contradiction. So P ≤ T. In this case, H = H ∩ T ∈ H(G) and then H G by the relationship H P G and Lemma 2(2), which is impossible. Thus, P is not a minimal normal of G.
(2) If every maximal subgroup of P is weakly H-embedded in G, then P ≤ Z U (G). Let N be a minimal normal subgroup of G contained in P. By Lemma 3(2), (G/N, P/N) satisfies the hypothesis. So, the choice of (G, P) implies that: (i) P/N ≤ Z U (G/N); (ii) N is non-cyclic; (iii) N is the unique minimal normal subgroup of G contained in P. Now assume that Φ(P) = 1. In this case, P is elementary abelian and P = N × B, where B is a complement of N. Let N 1 be a maximal subgroup of N such that N 1 is normal in some Sylow p-subgroup G p of G. Then P 1 = N 1 B is a maximal subgroup of P. By the hypothesis, G has a normal subgroup T such that . Hence Lemma 1(2) and the choice of N 1 imply that N 1 G, a contradiction. The above shows that Φ(P) = 1 and consequently, N ≤ Φ(P). Furthermore, P/Φ(P) ≤ Z U (G/Φ(P)). However, we have P ≤ Z U (G) by Lemma 4. This contradiction shows that (2) holds.
(3) If every cyclic subgroup of P of order p or 4 (when P is a non-abelian 2-group) is weakly H-embedded in G, then P ≤ Z U (G).
If P is not a non-abelian 2-group, then we use Ω to denote the subgroup Ω 1 (P) of P. Otherwise, Ω = Ω 2 (P).
Let R be a normal subgroup of G such that P/R is a G-chief factor. Obviously, R satisfies the hypothesis. So R ≤ Z U (G) and P/R is non-cyclic by the choice of (G, P). Moreover, for any normal subgroup L of G satisfying L < P, we have L ≤ R. In fact, if L R, then similarly L ≤ Z U (G), and P = RL ≤ Z U (G), a contradiction. Now, assume that Ω ≤ R. Then Ω ≤ Z U (G). From Lemma 4 and ([16] Lemma 2.4), it follows that G/C G (Ω) ∈ F(p) and C G (Ω)/C G (P) ∈ N p , where F is the canonical local satellite of U and N p is the class of p-groups. Consequently, G/C G (P) ∈ N p F(p) = F(p), and thereby P ≤ Z U (G) by Lemma 4 again. This contradiction shows that Ω = P.
Let L/R be a minimal subgroup of Z(G p /R) ∩ P/R and x ∈ L \ R, where G p is a Sylow p-subgroup of G. Then H = x has order p or 4 and L = HR. By the hypothesis, H is weakly H-embedded in G, so G has a normal subgroup T such that HT is S-quasinormal in G and H ∩ T ∈ H(G). Please note that P ∩ T G. Combining with the above result, we have P ∩ T = P or P ∩ T ≤ R. If P ∩ T = P, that is, P ≤ T, then H = H ∩ T ∈ H(G). Moreover, the relationship H P G and Lemma 2(2) deduce H G. By the choice of H, we have P/R = L/R is cyclic, which is a contradiction. Now assume that P ∩ T ≤ R. Then . From Lemma 1(2) and the choice of L/R, it follows that L/R G/R, which also shows that P/R = L/R, a contradiction. This completes the proof of (3).
(6) Final contradiction. Let N be a minimal normal subgroup of G contained in P. Clearly, N < P. Compare the order of N with |D|. If |D| < |N|, then N satisfies the hypothesis and the choice of P implies that N ≤ Z U (G). Consequently, |N| = p and then |D| = 1, which contradicts (4). Thus, |D| |N|. By (5), P is elementary abelian, and all subgroups of P/N of order |D|/|N| and p|D|/|N| are weakly H-embedded in G (see Lemma 3 (2)). Therefore P/N ≤ Z U (G/N) by the choice of P. Please note that |P/N| |P|/|D| > p 2 . So there exists a normal subgroup E of G contained in P satisfying N ≤ E ≤ P and |P/E| = p. Consider the subgroup E. Then E ≤ Z U (G) by the hypothesis and the choice of P, which implies |N| = p. Combining with P/N ≤ Z U (G/N), we finally obtain P ≤ Z U (G), which is a contradiction. The final contradiction completes the proof of the proposition.

Now we give the proof of Theorem 2:
Proof. Suppose that the assertion is false and consider a counterexample (G, E) with minimal |G| + |E|. We proceed via the following steps.
Suppose that E < G. Please note that Lemma 3(1) shows that (E, E) satisfies the hypothesis, so E is p-supersoluble. Combining (1) with Lemma 5, we have P E and consequently, P G. From the hypothesis and Proposition 1, it follows that P ≤ Z U (G). This result implies E ≤ Z pU (G) and then G is p-supersoluble, which is a contradiction. Thus, E = G.
(3) If every maximal subgroup of P is weakly H-embedded in G, then G is p-supersoluble. Let N be a minimal normal subgroup of G. Since G is p-soluble and O p (G) = 1, N ≤ O p (G). By Lemma 3(2), G/N satisfies the hypothesis, so: In fact, if P G, then P ≤ Z U (G) by Proposition 1. Similar to step (2), it is impossible.
Using the above symbol, G = O p (G) M and then P = O p (G) (P ∩ M). Let P 1 be a maximal subgroup of P containing P ∩ M.
by the minimality of O p (G) and consequently, P = P 1 , a contradiction. By the hypothesis, P 1 is weakly H-embedded in G. So G has a normal subgroup T such that P 1 T is S-quasinormal in G and P 1 ∩ T ∈ H(G). If T = 1, then P 1 is S-quasinormal in G, which implies that P 1 ≤ O p (G) by Lemma 1(3)(4) and then O p (G) = P. However, it contradicts the above result. So, the uniqueness of O p (G) implies that O p (G) ≤ T. Next, we prove that First, we show that N G (P 1 ∩ O p (G)) = N G (P 1 ∩ T). On one hand, note that On the other hand, N G (P 1 ∩ O p (G)) is p-supersoluble by Lemma 3(1) and the relation At this moment, we have and by Lemma 2(1), Together with the above proof, we finally obtain N G (P 1 ∩ O p (G)) = N G (P 1 ∩ T). Please note that P 1 ∩ T ∈ H(G). So, for any element g ∈ G, This shows that P 1 ∩ O p (G) ∈ H(G). By Lemma 2(2), we further have P 1 ∩ O p (G) G, a contradiction. This completes the proof of (3).
(4) If every cyclic subgroup of P of order p or 4 (when P is a non-abelian 2-group) is weakly H-embedded in G, then G is p-supersoluble.
Let M be any proper subgroup of G and M p a Sylow p-subgroup of M. Clearly, (M p ) g ≤ P for some element g ∈ G. Then consider M g , which has a Sylow p-subgroup (M p ) g contained in P. So, without loss of generality, assume that M p ≤ P. By Lemma 3(1), M satisfies the hypothesis, so the choice of G implies that M is p-supersoluble. As a result, G is a minimal non-p-supersoluble group.
By ( [17] Theorem 1), G U p Φ(G)/Φ(G) is the unique minimal normal subgroup of G/Φ(G), where U p is the class of all p-supersoluble groups. Clearly, p | |G U p Φ(G)/Φ(G)|, so G U p Φ(G)/Φ(G) is a p-group and G U p is solvable. From ([18] Theorem 3.4.2), it follows that G U p is a p-group of exponent p or 4 (when G U p is a non-abelian 2-group). By the hypothesis, every cyclic subgroup of G U p of order p is weakly H-embedded in G. When G U p is a non-abelian 2-group, clearly, P is also a non-abelian 2-group, so every cyclic subgroup of G U p of order 4 is also weakly H-embedded in G in this case. Hence, we have G U p ≤ Z U (G) by Proposition 1, and then G is p-supersoluble, a contradiction. So (4) holds.
by Proposition 1, which shows that |N| = p and |D| = 1, a contradiction. So, we have |N| |D|. Please note that p > 2, so it is easy to show that (G/N, P/N) satisfies the hypothesis. Thus, the choice of G implies that: G/N is p-supersoluble; |N| > p; N is the unique minimal normal subgroup of Then |P : P ∩ R| = p and by (6), R satisfies the hypothesis of the theorem. So R is p-supersoluble. Please note that O p (R) ≤ O p (G) = 1. Together with Lemma 5, R has the unique Sylow p-subgroup P ∩ R, and furthermore, P ∩ R G. By (6), P ∩ R satisfies the hypothesis of Proposition 1. Thus, P ∩ R ≤ Z U (G), that is, R ≤ Z pU (G), which deduces that G is p-supersoluble, a contradiction. Then assume that p |G/R|, that is P ≤ R. In this case, R satisfies the hypothesis and so R is p-supersoluble by the choice of G. Similarly, we have O p (R) = 1 and by Lemma 5, P R, which implies that P G. By Proposition 1, P ≤ Z U (G) and consequently, G is p-supersoluble, a contradiction. The final contradiction completes the proof of the theorem.

Next we give the proof of Theorem 3:
Proof. Suppose that the assertion is false and consider a counterexample G of minimal order. According to Theorem 1, we only need to consider that p is odd. We proceed via the following steps. ( . Please note that P a Sylow p-subgroup of E and N G (P) = N G (P) is p-nilpotent. Moreover, by hypothesis and Lemma 3(3), all subgroups of P of order |D| and order p|D| are weakly H-embedded in G, that is G satisfies the hypothesis for G. Thus, the choice of G implies that G is p-nilpotent. Consequently, G is p-nilpotent, a contradiction. So O p (E) = 1.
(2) E = G. By Lemma 3(1), all subgroups of P of order |D| and order p|D| are weakly H-embedded in E. Since N E (P) = N G (P) ∩ E, N E (P) is p-nilpotent. Then E satisfies the hypothesis. If E < G, then E is p-nilpotent by the choice of G. Let E p be the normal p -Hall subgroup of E. Clearly, E p G. So, by (1), E p = 1, that is, E = P. In this case, G = N G (P) is p-nilpotent. This contradiction shows that E = G.
(4) G is not p-soluble. Suppose that G is p-soluble. Then G is p-supersoluble by the Theorem 2. Please note that O p (G) = 1. So P G by Lemma 5, which shows that N G (P) = G is p-nilpotent, a contradiction. Thus, (4) holds. Consequently, G is a minimal non-p-nilpotent group. However, in this case, G is soluble, which contradicts (4). Suppose that |N| < |D|. Then all subgroups of P/N of order |D|/|N| and p|D|/|N| are weakly H-embedded in G/N by Lemma 3 (2), that is G/N satisfies the hypothesis for G. SoSo, from the choice of G, we deduce that G/N is p-nilpotent. Similarly, G is p-soluble in this case, a contradiction. Thus, |N| > |D|. (6) Final contradiction. By (5), all subgroups of N of order |D| and p|D| are weakly H-embedded in G. Then N ≤ Z U (G) by Proposition 1. From this result, we deduce that |N| = p and |D| = 1, that is, every subgroup of P of order p is weakly H-embedded in G. Similarly, as the proof of (5), we can prove that in this case G is soluble, a contradiction. The final contradiction completes the proof.

Some Applications
In this section, we list some applications of our results. Corollary 1. Let E be a normal subgroup of G. For every non-cyclic Sylow subgroup P of E, assume that P has a subgroup D such that 1 < |D| < |P| and all subgroups of P of order |D| and p|D| are weakly H-embedded in G. Then E ≤ Z U (G).
Proof. Assume that p is the smallest prime divisor of |E| and P is a Sylow p-subgroup of E. If P is cyclic, then E is p-nilpotent by the famous Burnside Theorem. Otherwise, by Lemma 3(1) and the hypothesis, all subgroups of P of order |D| and p|D| are weakly H-embedded in E. So E is p-nilpotent by Theorem 1, and then E is soluble. By Lemma 3(1) again, we have that for any prime p dividing |E|, E satisfies the hypothesis of Theorem 2. So E is supersoluble. Let q be the maximal prime dividing |E| and Q the unique Sylow q-subgroup of E. Clearly, Q G. Note that Q satisfies the hypothesis of Proposition 1, so Q ≤ Z U (G). Now consider E/Q. By Lemma 3(3), E/Q satisfies the hypothesis of corollary. So E/Q ≤ Z U (E/G) by induction. Therefore, E ≤ Z U (G). Corollary 2. ( [12]) Assume that the Sylow subgroups of G are non-cyclic for all primes p dividing |G|. Assume further that for each such p there is a p-power d with 1 < d < |G| p such that all subgroups of P of order d and pd are weakly H-embedded in G, then G is supersoluble.
Proof. Let p be the smallest prime dividing |G|. By Theorem 1, G is p-nilpotent. Consequently, G is soluble. From the Theorem 2, it follows that G is q-supersoluble, for any prime divisor q of |G|, that is, G is supersoluble.  (1) G has a normal subgroup H such that G/H is supersolvable and all maximal subgroups of every Sylow subgroup of H belong to H(G) [7]; (2) all maximal subgroups of every Sylow subgroup of F * (G) belong to H(G) [7]; (3) all maximal subgroups of every Sylow subgroup of a group G are weakly H-subgroups in G [10].

Conclusions
In this paper, we further explore weakly H-embedded subgroups. As new applications, we generalize the characterization of p-nilpotent given by Asaad, Ramadan and Heliel and get a new criterion for p-supersolubility for general prime p. Moreover, adding condition "N G (P) is p-nilpotent", we obtain p-nilpotence for general prime p.