Influence of partially τ-embedded subgroups of prime power order in supersolubility and p-nilpotency of finite groups

Abstract In this paper, we introduced a new concept of embedded subgroups which belongs to an embedded class of subgroups of finite groups. A subgroup H of a group G is said to be a partially τ-embedded subgroup in G if there exists a normal subgroup K of G such that HK is normal in G and where generated by all those subgroups of H which are partially τ-quasinormal in G. We investigate the influence of some -embedded subgroups with prime power order on the structure of a finite group G. Some new criteria about the p-nilpotency and supersolubility of a finite group were obtained. Our results also generalized some earlier ones about formations.


Introduction
During the twentieth century, mathematicians investigated some aspects of the theory of finite groups in great depth, especially the local theory of finite groups and the theory of solvable and nilpotent groups. As a consequence, the complete classification of finite simple groups was achieved, meaning that all those simple groups from which all finite groups can be built are now known. During the second half of the twentieth century, mathematicians such as Chevalley and Steinberg also increased our understanding of finite analogs of classical groups, and other related groups. One such family of groups is the family of general linear groups over finite fields. Finite group theory fund many applications in chemistry, physics, engineering and other areas of sciences.
In this article, G denotes the finite groups. All the notations are standard, as in [1,2]. Order of G denoted by |G|, Sylow q-subgroup of G denoted by G q and Sylow subgroup of G denoted by Syl(G) or simply Syl. partially τ -quasinormal subgroups denoted by pτ -quasinormal and partially τ -embedded subgroups simply denoted by pτ -embedded.
Many authors worked on quasinormal subgroups and gave generalizations of normal subgroups. For example, Kegel [3] gave the extension of S-quasinormal. τ -quasinormality is the generalization of the S-quasinormal subgroups of finite groups. Recently, Li et al. [2] generalized τ -quasinormal subgroups to pτ -quasinormal. H is pτ -quasinormal in G if every Syl(H) is a Sylow subgroup of some τ -quasinormal subgroup of G. Many authors discussed the concept of permutable groups and gave many new concepts and generalizations of permutable groups. Now we present another extension of permutable groups and by the use of the following definitions we extend the permutable groups.

Theorem 1.4: Let H
G such that G/H is supersolvable. If every maximal subgroup of any Syl of (H) is pτ -embedded in G, then G is supersolvable. Theorem 1.5: Let Q be a G q , where q is a prime divisor of |G| with (|G|, (q − 1)(q 2 − 1) · · · (q n − 1)) = 1 (n ≥ 1). If every maximal subgroup of Q and every n-maximal subgroup of Q is pτ -embedded in G, then G is q-nilpotent. Theorem 1.6: Suppose M G and Q be a Sylow q-subgroup of M such that (|M|, q − 1) = 1 provided q is a prime divisor of |M|. If every largest subgroup Q 1 of Q is pτ -embedded in G such that Q 1 does not have a qsupersolvable supplement in G, then each chief factor of G between M and O q (M) is cyclic.

Preliminaries
This section contains some basic results that help us in proving our main results.

Lemma 2.1:
Suppose that A and B are two subgroups of a finite group G, then the following statements hold:

Lemma 2.3 ([5]):
Let Q is S-quasinormal q-subgroup of the finite group G for some prime q, then we have  Proof: Suppose that L be a partially τ -quasinormal subgroup of G contained in H and q be a prime dividing |L|. Further suppose that Q a Sylow q-subgroup of L and E an S-quasinormal subgroup of G such that Q ∈ yl q (E). Then the proof is given as follows: (1) Let x ∈ H, then L x ≤ H. If R is a Sylow q-subgroup of L x , then R = Q x 1 for some Sylow q-subgroup Q 1 = Q. Obviously, Q x ∈ Syl q (E x ) and E x is an S-quasinormal subgroup of G. Hence L x is a partially τ -quasinormal subgroup of G. This implies that (1) holds.
(2) By Lemma 2.1(1), E ∩ K is an s-quasinormal subgroup of K. Since Q ≤ E ∩ K, Q is a Sylow q-subgroup of E ∩ K. Hence L ≤ H pτ K and so H pτ G ≤ H pτ K . (  Hence ,

Lemma 2.6 ([7]):
Consider a group G and prime number q such that q n+1 does not divides |G| for integers n ≥ 1. If then G is q-nilpotent.

Lemma 2.8 ([9]):
Suppose that P, R, S ≤ G. Then the following statements are equivalent: (2) If G has G q a cyclic subgroup, then G is q-nilpotent.
Lemma 2.11: Let q be a prime divisor of the order of G in such a way (|G|, q − 1) = 1, then (2) G is q-nilpotent provided G has cyclic Sylow q-subgroup.
(3) X is normal in G provided |G : X| = q and X ≤ G.

(4) N lies in Z(G) provided |N| = q and N is normal in G.
Proof: Suppose C/D is any random chief factor. If G is q-supersolvable, then there are two possibilities: (1) |C/D| = q is cyclic. ( As Hence G is q-nilpotent. Proof of (2), (3) and (4)

Lemma 2.13: Suppose X G is q-subgroup. Then there exists A G, such that A is the largest subgroup of X and is pτ -embedded.
Proof: If the order of X is q, then the theorem holds. Let Y ≤ X be a normal q-subgroup which is smallest without identity. Let X = Y. Using Lemma 2.6(2) of [13], the theorem is still satisfied by G/Y. So with the help of induction some largest subgroup Obviously, L ≤ X and L G. Therefore, the hypothesis is satisfied again. Now let X = Y. Assume that L be any largest subgroup of X. So there will be E G in such a way that LE is S-quasinormal Hence X ≤ E, which shows that So using 2.3 of [13], LE ∩ X is S-quasinormal, which is again a contradiction. Thus, L = L pτ G . So using 2.3 of [13], L is S-quasinormal. Consequently, X G by Lemma 2.11 of [14]. Hence the lemma is proved.

Proofs of main results
In this section, we give proofs of our main theorems.

Proof of Theorem 1.4:
The proof of this follows in the following steps: (1) In this step, we show that (G) ≤ G is supersolvable.
From (4) each maximal of Syl of F(G) is pτembedded, so from Theorem 4.3 of [10]. G is supersolvable.
This completes the proof of Theorem 1.4.

Proof of Theorem 1.5:
The proof of this theorem is given in the following steps (1) Using Lemma 2.6, | Q | ≥ q n+1 ; thus every n-maximal subgroup Q n of Q satisfies Q n =1.
(2) Now we prove that G is not simple.
By hypothesis, Q n is pτ -embedded. Using the definition of the pτ -embedded subgroup and K G s.t. Q n K G, Q n ∩ K ≤ (Q n ) pτ G . Let G be simple. If K = 1, then 1 = Q n K = Q n G, which contradicts the hypothesis. If K = G, then 1 < Q n ∩ K = Q n ≤ (Q n ) pτ G . We can write Let V be an arbitrary nontrivial pτ -quasinormal subgroup ≤ Q n . Then T ≤ G be S-quasinormal subgroup such that V be Syl q (T). As G be a simple group, we have T G = 1, By Lemma 2.1, V is S-quasinormal. From the arbitrariness of V and Lemma 2.1 of [2], Q n is S-quasinormal, so Q n = 1, in contrary to (1). (3) Now we prove that M G, where M is unique and minimal.
(G) is equal to 1. Since G/M satisfies which shows that QM/M is a Syl q (G/M). By Lemma 2.6, we may take |QM/M| ≥ q n+1 . Let N n /M be n-maximal of QM/M. So N n = N n ∩ QM = (N n ∩Q)M = Q n M. Obviously, Q n is an n-maximal subgroup of Q. According to supposition, Q n is pτ -embedded. Therefore, there is K G such that Q n K G and Consequently, Q = MQ 1 = Q 1 , which contradicts the hypothesis. Thus we have T G = 1. Furthermore, using Lemma 2.2, V is τ -quasinormal. From the arbitrariness of V and Lemma 2.1 of [2], (Q n ) pτ (G) is τ -quasinormal. By Lemmas 2.3 of this paper and Lemma 2.1 of [2], O q (G) ≤ M G ((Q n ) pτ (G) ) and (Q n ) pτ (G) is subnormal in G. By Lemma 2.7, pτ (G) = 1, then Q 1 ∩ K = 1 and so K q ≤ q n . So K is q-nilpotent from Lemma 2.6. Suppose K q is normal q-complement of K, then K q G, we get K q = 1 by step (4), and thus there is q-subgroup K G and K ≤ Q n K ≤ O q (G) = M. If K = 1, we get K = Q n K = M, so Q n ≤ K, namely, Q n ∩ K = Q n = 1, which contradicts the hypothesis. If K = 1, then Q n G, so M ≤ Q n ≤ Q 1 , which contradicts the hypothesis. Now it is clear that (5) holds. (6) Now we complete the proof with the following lines: If M ∩ Q ≤ (Q), then M is q-nilpotent by Tate s Theorem [19,IV,4.7]. Therefore, M q G. So M q ≤ O q (G) = 1. Moreover, M be q-group, then M ≤ O q (G) = 1, which contradicts the hypothesis. As a result, there is a maximal Q 1 ≤ Q, s.t. Q = (Q ∩ M)Q 1 . Take Q n ≤ Q contained in Q 1 . By the hypothesis, K is normal in G s.t. Q n K G, Q n ∩K ≤ (Q M ) pτ G . Let V be a nontrivial pτ -quasinormal contained in Q n . So τ -quasinormal T ≤ G, then V be Syl q (T). If which contradicts the hypothesis. Hence T G = 1, V is τ -quasinormal from Lemma 2.1. By Lemma 2.1 of [2] and arbitrariness of V, (Q M ) pτ G is S-quasinormal, and so (Q M ) pτ G is subnormal using Lemma 2.1 of [2]. By Lemma 2.7 that (Q M ) pτ G ≤ O q (G) = 1, so |K q | ≤ q n , therefore K is q-nilpotent. Similarly, we have K q = 1 and so K = 1. It deduces that Q n G, M ≤ Q n ≤ Q 1 .
This completes the proof of Theorem 1.5.

Proof of Theorem 1.6:
Here we will prove the theorem by obtaining a contradiction. The proof follows in the following steps: (1) First, we prove that K is q-nilpotent.
Let Q 1 be the largest subgroup of Q. Q 1 has a qsupersolvable supplement X ∩ K in K provided Q 1 has q-supersolvable supplement X. Because (|K|, q − 1) = 1, this implies X ∩ K is q-nilpotent from Lemma 2.11 (1). If Q 1 is pτ -embedded in G, so Q 1 is also pτ -embedded in K from 2.6(1) of [13]. Also, Q 1 does not have any q-nilpotent supplement in K. So by theorem 1.5 of [13], K is q-nilpotent. (2) Now we prove that Q = K.
Using step (1), O q (K) is the normal Hall q -subgroup of K. Let O q (K) = 1. We can check it easily that our theorem is true for (G/O q (K, K/O q (K)). Using mathematical induction we can see G/O q (K) be the chief factor, between 1 and K/O q (K)) is cyclic. Following each factor between K and O q (K) is cyclic, which implies O q (K) = 1. Hence Q = K. (3) In this step, we prove that (Q) = 1.
First, we let (Q) = 1. Then in the light of Lemma 2.3(2), we can check easily that our theorem holds for (G/ (Q), Q/ (Q)). Every chief factor of G/ (Q) under Q/ (Q) is cyclic by our selection of (G,K). Hence cyclic by Lemma 2.12, which contradicts the hypothesis. (4) Here we prove that every largest subgroup of Q is pτ -embedded. Let us have some largest Q 1 subgroup contained in Q in such a way that T is the q-supersolvable supplement of Q 1 in G, thus QT = G with Q ∩ T = 1. Because Q ∩ T T, we may suppose that Q ∩ T contains a smallest normal subgroup L of T. Here Obviously |L| = q. Since Q is elementary abelian and G = QT, this implies L G. Here we can check that our theorem holds yet for (G/L, Q/L). By our selection of (G,K) we can see every chief factor of G/L under Q/L is cyclic. As a consequence, every chief factor of G under Q is cyclic, which is a contradiction, hence (4) holds. (5) Now we find the smallest normal subgroup.
Let Q G, so using Lemma 2.13, G contains some largest normal subgroup of Q, which can't be true because Q is of smallest order. (6) Let L Q of G, then Q/L ≤ N V (G/L), and |Y| > q.
Moreover, using Lemma 2.3(2) of [13], our theorem satisfies (G/L, Q/L). Thus from our selection of (G, K) = (G, Q), every chief factor of G/L under Q/L is cyclic. If |L| = q, then cyclic, which contradiction of our supposition. Now if Q contains two smallest normal subgroups R and L of G, then LR/R ≤ Q/R and from the isomorphism LR/R ∼ = L, it follows that |L| = q, a contradiction again. Thus, step (6) is true. (7) Finally, we prove the contradiction.
Suppose that L Q of G and L 1 the largest subgroup of L. To show L 1 is S-quasinormal. So we may suppose that B is a complement of L in Q, as Q is an elementary abelian q-group. Also take W = L 1 B. Clearly, W is a largest subgroup of Q. Using step (4), W is pτ -embedded in G. So using Lemma 2.6(4) of [13], there will be R G satisfying the condition, W ∩ R ≤ W pτ G , WR ≤ Q and WR is S-quasinormal. From Lemma 2.3 of [13], W pτ G is S-quasinormal. Now if R = Q, so W = W pτ G is S-quasinormal. By 2.1(5) of [2], is S-permutable. If R = 1, this gives W = WR is S-quasinormal. As a result, L 1 is S-quasinormal. Consider 1 < R < Q. Implies L ≤ R by step (6). So by using Lemma 2.1(5) of [2], is S-quasinormal. This implies |L| = q, which contradicts step (6).

Conclusions
In this paper, we check the supersolvability and nilpotency of pτ -embedded subgroups. We proved that if H G such that G/H is supersolvable and every maximal subgroup of any Syl of (H) is pτ -embedded in G, then G is supersolvable. Further we proved that if Q be a G q , where q is a prime divisor of |G| with ( |G|, (q − 1)(q 2 − 1) · · · (q n − 1)) = 1 (n ≥ 1) and every maximal subgroup of Q and every n-maximal subgroup of Q is pτ -embedded in G, then G is q-nilpotent. At last, we prove that if M G and Q be a Sylow q-subgroup of M such that ( |M|, q − 1) = 1 provided q is a prime divisor of |M| and every largest subgroup Q 1 of Q is pτ -embedded in G such that Q 1 does not have a qsupersolvable supplement in G, then each chief factor of G between M and O q (M) is cyclic. Our results are the extension of existing results.

Disclosure statement
No potential conflict of interest was reported by the authors.