ON NH -EMBEDDED AND SS -QUASINORMAL SUBGROUPS OF FINITE GROUPS

. Let G be a finite group. A subgroup H is called S -semipermutable in G if HG p = G p H for any G p ∈ Syl p ( G ) with ( | H | ,p ) = 1, where p is a prime number divisible | G | . Furthermore, H is said to be NH -embedded in G if there exists a normal subgroup T of G such that HT is a Hall subgroup of G and H ∩ T ≤ H sG , where H sG is the largest S -semipermutable subgroup of G contained in H , and H is said to be SS -quasinormal in G provided there is a supplement B of H to G such that H permutes with every Sylow subgroup of B . In this paper, we obtain some criteria for p -nilpotency and Supersolvabil-ity of a finite group and extend some known results concerning NH -embedded and SS -quasinormal subgroups.


Introduction
Throughout, all groups considered in this paper will be finite. Let G be a group, H and K of are subgroups of G, they are said to be permutable if HK = KH, i.e.
HK is also a subgroup of G. H is a subgroup of G, it is said to be quasinormal in G if H permutes with all subgroups of G. Kegel [8] introduced the concept of S-quasinormal (or S-permutable), subgroup H of G said to be S-quasinormal if H permutes with all Sylow subgroup of G. Recall that a supplement of H to G is a subgroup B such that G = HB. As a generalization of S-quasinormal subgroup, Li [9] introduced the following definition: Definition 1.1. [9] A subgroup H of G is said to be SS-quasinormal in G provided there is a supplement B of H to G such that H permutes with every Sylow subgroup of B.
Li [9] investigated the p-nilpotency and supersolvability of finite groups by some SS-quasinormal subgroups of prime power order.
Recall that a subgroup H is called S-semipermutable in G if HG p = G p H for any G p ∈ Syl p (G) with (|H|, p) = 1, p is a prime number divisible G(see [2]). Recently, Gao and Li [5] introduce the following concept: Definition 1.2. [5] A subgroup H of a group G is said to be N H-embedded in G if there exists a normal subgroup T of G such that HT is a Hall subgroup of G and Gao and Li [5] showed that the finite group whose maximal subgroups of Sylow subgroups are N H-embedded in G are supersolvable.
By the definition of N H-embedded and SS-quasinormal subgroups, it is obvious that all Hall subgroups, normal subgroups and S-semipermutable subgroups are N H-embedded subgroups. But the converse does not hold. Moreover, an N Hembedded subgroup need not be SS-quasinormal. Conversely, it easy to see that an SS-quasinormal subgroup need not be N H-embedded too.
In the light of above results, it is seem interesting to study the structure of finite groups assuming that maximal subgroups of Sylow subgroups are SS-quasinormal or N H-embedded in G. In this paper, we obtain some criteria for p-nilpotency and supersolvability of a finite group. The main results are as follows. Theorem 1.3. Let G be a finite group and G p a Sylow p-subgroup of G, where p is the smallest prime dividing |G|. Assume that every maximal subgroup of G p is either N H-embedded or SS-quasinormal in G. Then G is p-nilpotent.  All unexplained notations and terminologies are standard and can be found in [4,6].

Preliminaries
In this section, we collect some results which will be used in the proof of main results.
Lemma 2.1. [9] Suppose that H is SS-quasinormal in a group G, K ≤ G, and N is a normal subgroup of G. We have (2) HN/N is SS-quasinormal in G/N ; (1) H is S-quasinormal in G;  (1) H is an S-quasinormal subgroup of G; (2) the Sylow subgroups of H are S-quasinormal in G.
Lemma 2.6. [1] Let P be a Sylow p-subgroup of a group G, and let P 0 be a maximal subgroup of P . Then the following statements are equivalent: (1) P 0 is normal in G; (2) P 0 is S-quasinormal in G.
[10] Let G be a group and H an S-semipermutable subgroup of G.
Suppose that H is a p-subgroup of G for some prime p ∈ π(G) and N is normal in Lemma 2.10. [7] Let H be an S-semipermutable π-subgroup of G. Then H G contains a nilpotent π-complement, and all π-complements in H G are conjugate.
Also, if π consists of a single prime, then H G is solvable.

Proof of Theorem
Proof of Theorem 1.3 Assume that the theorem is false and let G be a counterexample of minimal order. Let G p be a Sylow p-subgroup of G and M(G p ) = {P 1 , P 2 , · · · , P m } denote the set of all maximal subgroups of G p . By Theorem hypothesis, every member P i of M(G p ) is either N H-embedded or SS-quasinormal in G. Without loss of generality, suppose that every member of the subset M 1 (G p ) = {P 1 , · · · , P k } of M(G p ) is N H-embedded in G, and every member of the subset The proof of theorem will be divided into five steps as follows.
Step (1). G has a unique minimal normal subgroup N and G/N is p-nilpotent.
Theorem hypothesis, P is either N H-embedded or SS-quasinormal in G.

5
Suppose that P is SS-quasinormal in G, then M/N = P N/N is also SSquasinormal in G/N by Lemma 2.1 (2). Now, we assume that P is N H-embedded in G, then there is a normal subgroup T of G such that P T is a Hall subgroup of G and P ∩ T ≤ P sG . It is easy seen that T N/N is normal in G/N and P N/N · T N/N = P T N/N is a Hall subgroup of This implies that N ∩ P T = (N ∩ P )(N ∩ T ) and hence P N ∩ T N = (P ∩ T )N by Lemma 2.11. Therefore, As P sG is S-semipermuted in G, we get that P sG N/N is also S-semipermuted in G/N by Lemma 2.8 (2). This leads to By above arguments, we know that G/N satisfies the hypothesis of theorem. By the choice of G, we know that G/N is p-nilpotent. Moreover, as the class of all p-nilpotent groups is saturated formation, we obtain that N is the unique minimal normal subgroup of G.
Suppose that O p ′ (G) > 1, then N ≤ O p ′ (G) by (1). As G/N is p-nilpotent, we know that G/O p ′ (G) is p-nilpotent and hence G is also p-nilpotent, a contradiction.
Step (3). N ≤ P i for any P i ∈ M 1 (G p ).
For any H ∈ M 1 (G p ), H is N H-embedded in G, then there is a normal subgroup T of G such that HT is a Hall subgroup of G and H ∩ T ≤ H sG . If T = 1, then H is a Hall subgroup of G and hence H = 1. This implies that |G p | = p, as P is a maximal subgroup of G p . By Burnside theorem, G is p-nilpotent, a contradiction.
Burnside theorem. Let U be a normal Hall p ′ -subgroup of N , then U is normal in G. By minimality of N , we know that U = 1 and hence N is a subgroup of order p.
Consequently, the nilpotency of G/N implies that G is p-nilpotent, a contradiction.
Step (4). For every P j ∈ M 2 (G p ), there exists a normal subgroup M j of G such For any H ∈ M 2 (G p ), we know that H is SS-quasinormal in G. So there exists a subgroup B of G such that G = HB, and HB p = B p H for every Sylow subgroup  Step (5). Final contradiction.
M j ). By above arguments, we know that N ≤ V .

Moreover, we have
By ( [6]. III. 3.3), we know that Φ(G p ) ≤ Φ(G) and hence N ≤ Φ(G). Since G/N is p-nilpotent, we get that G/Φ(G) is p-nilpotent. As the class of all p-nilpotent is a statured formation, G will be p-nilpotent. This is finally contradiction. The proof of theorem is complete. □ Proof of Theorem 1.4 Assume that the theorem is false and let G be a counterexample of minimal order. Let p be the smallest prime dividing |G|. Then G is p-nilpotent by Theorem 1.3. Let U be a Hall normal p ′ -subgroup of G. It is easy seen that U satisfies the theorem hypothesis by Lemma 2.1(1) and Lemma 2.7(1).
By induction, U is supersolvable and hence G possesses Sylow tower property of supersolvable type. Let q be the largest prime dividing |G| and Q is a Sylow qsubgroup of G. Then Q is normal in G. By Lemmas 2.1(2) and 2.7(3), we know that G/Q satisfies the theorem hypothesis and hence G/Q is supersolvable by the choice of G.
Let N be a minimal normal subgroup of G. Similar to the proof of Theorem 1.3, G/N satisfies the theorem hypothesis and hence G/N is supersolvable by minimality of G. As the class of all supersolvable group is a statured formation, N will be a unique minimal normal subgroup of G. Therefore, we have N ≤ Q.
We claim that N ≤ H for any H ∈ M(Q). By theorem hypothesis, we know As G/Q is supersolvable, we get that G would be supersolvable, a contradiction. Therefore, T ̸ = 1 and hence N ≤ T . Consequently, G/T is supersolvable. If Let N be a minimal normal subgroup of G contained in Q. Similar to the proof of Theorem 1.3, we know that G/N satisfies the theorem hypothesis and hence G/N is supersolvable. As the class of all supersolvable group is a statured formation, N will be a unique minimal normal subgroup of G contained in Q.
In the following, similar to the proof of Theorem 1.4, we can get that N ≤ Φ(Q) and hence G/Φ(Q) is supersolvable. By ( [6], III, 3.3), Φ(Q) ≤ Φ(G). So G/Φ(G) is supersolvable. As the class of all supersolvable groups is a statured formation, we obtain that G is supersolvable. This is a final contradiction. The proof of Theorem is complete. □

Some applications
As an immediate consequence of Theorem 1.3, we can get the corollaries as follows.