Finite Groups with Given Systems of m-S-Complemented Subgroups

Let G be a finite group and H a subgroup of G. We say that H: is generalized S-quasinormal in G if H=A,B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H=\left\langle A,B\right\rangle $$\end{document} for some modular subgroup A and S-quasinormal subgroup B of G; m-S-complemented in G if there are a generalized S-quasinormal subgroup S and a subgroup T of G such that G=HT\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G=HT$$\end{document} and H∩T≤S≤H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H\cap T\le S\le H$$\end{document}. In this paper, we study finite groups with given systems of m-S-complemented subgroups. In particular, we prove that if F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak {F}}$$\end{document} is a saturated formation containing all supersoluble groups and E is a normal subgroup of a finite group G such that G/E∈F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G/E\in {\mathfrak {F}}$$\end{document} and for every non-cyclic Sylow subgroup P of E every maximal subgroup of P not having a nilpotent supplement in G is m-S-complemented in G, then G∈F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ G\in {\mathfrak {F}}$$\end{document} .

A subgroup M of G is called modular in G [1, p. 43] if (1) X , M ∩ Z = X , M ∩Z for all X ≤ G, Z ≤ G such that X ≤ Z , and (2) M, Y ∩ Z = M, Y ∩ Z for all Y ≤ G, Z ≤ G such that M ≤ Z .
A subgroup H of G is said to be S-permutable [2,3] or S-quasinormal [4] in G if H permutes with every Sylow subgroup P of G, that is, H P = P H. The subgroup H of G is said to be generalized S-quasinormal in G [5] if there are a modular subgroup A and an S-quasinormal subgroup B of G such that H = A, B .
Interesting applications of generalized S-quasinormal subgroups were discussed in the paper [5]. In this paper, we consider the following generalization of such subgroups. Definition 1. 1 We say that a subgroup H of G is m-S-complemented in G if there are a generalized S-quasinormal subgroup S and a subgroup T of G such that G = H T and H ∩ T ≤ S ≤ H .
It is clear that every generalized S-quasinormal subgroup is m-S -complemented. Every modular subgroup and every S-quasinormal subgroup are generalized Squasinormal. Now consider the following Example 1.2 (1) Let C 3 A 4 = P A 4 , where A 4 is the alternating group of degree 4 and P is the base group of the regular wreath product C 3 A 4 . Let G = (P A 4 ) × (C 11 C 5 ), where C 11 C 5 is a non-abelian group of order 55. Let Q be the Sylow 2-subgroup of A 4 and R a Sylow 3-subgroup of A 4 . Then, P Q is supersoluble, so some subgroup B of P with |B| = 3 is normal in P Q. Then, for every Sylow 3-subgroup G 3  Next, we show that H is not generalized S-quasinormal in G. First note that H G = 1, so for every modular subgroup V of H we have V G ≤ C 11 C 5 by Lemma 2.4 below. Therefore, A is the largest modular subgroup of H . Assume that H is generalized Squasinormal in G and let W be an S-quasinormal subgroup of G such that H = A, W = AW . Then, W G = 1, so W is a nilpotent subnormal subgroup of G by [2,Theorem 1.2.17]. Hence for a Sylow 2-subgroup (2) A subgroup H of G is said to be complemented (respectively, c-supplemented [6]) in G, if there is a subgroup T of G such that G = H T and H ∩T = 1 (respectively, G = H T and H ∩ T ≤ H G ). It is clear that every complemented subgroup and every c-supplemented subgroup are m-S-complemented.
(3) A subgroup H of G is said to be S-supplemented [7] (respectively, msupplemented [8]) in G, if there are an S-quasinormal subgroup (respectively, a modular subgroup) S and a subgroup T of G such that G = H T and H ∩ T ≤ S ≤ H . Every S-supplemented subgroup and every m-supplemented subgroup are m-Scomplemented.
Let K ≤ H be normal subgroups of G. Then we say, following [1] that H /K is hypercyclically embedded in G if every chief factor of G between H and K is cyclic. We say also that H is hypercyclically embedded in G if H /1 is hypercyclically embedded in G.
In this paper, we prove the following results in this line research.

Theorem 1.4 Let E be a normal subgroup of G. Suppose that for any Sylow subgroup P of E every maximal subgroup of P not having a nilpotent supplement in G is m-S-complemented in G. Then, E is hypercyclically embedded in G.
Recall that the formation F is a homomorph of groups such that each group G has the smallest normal subgroup (denoted by G F ) whose quotient is still in F. A formation F is said to be saturated if G ∈ F for any group G with G/ (G) ∈ F.
As a first application of Theorem 1.4, we prove also the following theorem which covers many known results (see Sect. 4 below). Theorem 1.5 Let F be a saturated formation containing all supersoluble groups, and let X ≤ E be normal subgroups of G with G/E ∈ F. Suppose that for any Sylow subgroup P of X every maximal subgroup of P not having a nilpotent supplement in G is m-S-complemented in G. If X = E or X = F * (E), then G ∈ F.
In this theorem, X = F * (E) denotes the generalized Fitting subgroup of E [18, Ch. X], that is, the product of all normal quasinilpotent subgroups of E.

Preliminaries
The first lemma collects the properties of S-quasinormal subgroups used in our proofs.

Lemma 2.1 (See Chapter 1 in [2]). Let A, B and N be subgroups of G, where A is S-quasinormal in G and N is normal in G.
(1) AN /N is S-quasinormal in G/N . (

Lemma 2.2 Let A, B and N be subgroups of G, where A is generalized S-quasinormal in G and N is normal in G. Then (1) AN /N is generalized S-quasinormal in G/N . (2) If A ≤ B, then A is generalized S-quasinormal in B. (3) If N ≤ B and B/N is generalized S-quasinormal in G/N , then B is generalized S-quasinormal in G. (4) If B is generalized S-quasinormal in G, then A, B is generalized S-quasinormal in G.
Proof Let A = L, T , where L is modular and T is S-quasinormal subgroups of G.

Lemma 2.3 Let A, B and N be subgroups of G, where A is m-S-complemented in G and N is normal in G.
(

then A is m-S-complemented in B. (3) If N ≤ B and B/N is m-S-complemented in G/N , then B is m-S-complemented in G.
Proof Let T be a subgroup of G such that AT = G and A ∩ T ≤ S ≤ A for some generalized S-quasinormal subgroup S of G. Then, S = L, M , where L is a modular and M is an S-quasinormal subgroups of G. The lemma is proved.

Lemma 2.5 (See Theorem 1.2 in [12]). If E is a normal subgroup of G and F * (E) is hypercyclically embedded in G, then E is hypercyclically embedded in G.
Lemma 2.6 (See Lemma 2.16 in [7]). Suppose that G/N ∈ F, where F is a saturated formation containing all supersoluble groups. If N is hypercyclically embedded in G, then G ∈ F. Lemma 2.7 (See Lemma 2.10 in [9]). Let P be a Sylow p-subgroup of G, where p is the smallest prime dividing |G|. If every maximal subgroup of P has a p-nilpotent supplement in G, then G is p-nilpotent. Lemma 2.8 (See Lemma 2.12 in [19]). Let P be a normal p-subgroup of G. If P/ (P) is hypercyclically embedded in G, then P is hypercyclically embedded in G.

Proofs of Theorems 1.3, 1.4 and 1.5
The product of all hypercyclically embedded subgroups of G is denoted by Z U (G) and it is called the supersoluble hypercentre of G. Note that if A and B are normal hypercyclically embedded subgroups of G, then (in view of the G-isomorphism AB/A B/(B ∩ A)) the product AB is also hypercyclically embedded in G.

Proof of Theorem 1.3. Suppose that this theorem is false and consider a counterexample
(1) If R is a minimal normal subgroup of G and R is either a p -group or a p-subgroup contained in E such that R = P, then the hypothesis holds for (G/R, E R/R). First, suppose that R is a p -group. Then, p = |P||V ∩ R| : |P ∩ R||V ∩ P| = |P : V ∩ P|, so V ∩ P is a maximal subgroup of P. Then, by hypothesis, either V ∩ P has a p-nilpotent supplement S in G or V ∩ P is m-S-complemented in G. In the first (1). Now suppose that R is a p-subgroup contained in E. Then, R ≤ P and so p = |(P R/R) : (V /R)| = |P : V |. Then, by hypothesis, either V has a p-nilpotent supplement S in G or V is m-S-complemented in G. Therefore, V /R has a p-nilpotent supplement S R/R in G/R or V /R is m-S-complemented in G/R by Lemma 2.3 (1). Hence, the hypothesis folds for (G/R, E R/R).
Since p is the smallest prime dividing |E| by hypothesis, this follows from the fact that E/C E (H /K ) V ≤ Aut(H /K ) and from the fact that Aut(A) is a cyclic group of order p − 1 for any group A of order p.
(3) If R is a minimal normal subgroup of G and R is either a p -group or a psubgroup contained in E such that R = P, then E R/R is p-nilpotent and Hence, E is p-nilpotent and from and from the G-isomorphisms we get that E/O p (E) is hypercyclically embedded in G, contrary to the choice of (G, E). Hence, we have (4).
Since Z is clearly supersoluble, a Sylow q-subgroup Q of Z , where q is the largest prime dividing |Z |, is normal and so characteristic in Z . Then, Q is normal in G, which implies that Z = Q and q = p by Claim (4), so Z ∩ E ≤ Z ∞ (E) ≤ P since p is the smallest prime dividing E by hypothesis. (6) P = R for each minimal normal subgroup R of G.
Assume that P = R and let V be any maximal subgroup of P. Then, by hypothesis, either V has a p-nilpotent supplement S in G or V is m-S-complemented in G.
In the former case, we have S = G since G is not p-nilpotent. On the other hand, in this case, we have P = V (P ∩ T ) , where P ∩ T is clearly normal in G and so the minimality of R = P implies that P ∩ T = 1. But then V = P. This contradiction shows that V is m-S-complemented in G, so there are an m-S-permutable subgroup S and a subgroup T of G such that G = V T and V ∩ T ≤ S ≤ V . Let A be a modular subgroup and B an S-quasinormal subgroup of G such that S = A, B . Then, A G = 1, so A G ≤ Z by Lemma 2.4. Therefore, A = 1 and so S = B is S-quasinormal in G. But then S is normal in G by Lemma 1.2.16 in [2]. Hence, S = 1 and so T ∩ V = 1 . But then 1 < T ∩ R < R, where T ∩ R is normal in G. This contradiction shows that we have (6). Therefore, E is supersoluble, which implies that a Sylow q-subgroup Q of E, where q is the largest prime dividing |E|, is normal and hence characteristic in E. Hence, q = p and E = P = Q by Claim (4).
Assume that this is false. Then, E = P, so E = G by Claim (7). Since Z ∞ (G) is nilpotent, a Sylow p-subgroup of Z ∞ (G) is normal in G, so A = S = 1 since V G = 1. Therefore, T is a complement to V in G, so for a Sylow p-subgroup T p of T we have |T p | = p . Therefore, T is p-nilpotent by [20,IV,2.8]. Hence, every maximal subgroup V of P has a p-nilpotent complement in G, so G is p-nilpotent by Lemma 2.7. This contradiction shows that we have (a). (3) and (6). Hence, G is p-soluble. Therefore, every minimal normal subgroup R of G is a p-group by Claim (2). Hence, R is the unique minimal normal subgroup of G and R (G), so R = C G (R) = O p (G) by [33,Ch. A,15.6]. It is clear also that |R| > p, so Z = 1.
Let V be any maximal subgroup of P. We show that V has a p-nilpotent supplement in G. Assume that this is false. Then, the subgroup V is m-S-complemented in G by hypothesis.
First suppose that R V . Then, W = V ∩ R is normal in P, |R : W | = p and, by Claim (b), V G = 1. There are an generalized S-quasinormal subgroup S and a subgroup T of G such that G = V T and V ∩ T ≤ S ≤ V . Then, V ∩ T = S ∩ T . Arguing as above, we can show that S is S-quasinormal in G. Hence, S is subnormal in G by Lemma 2.1 (4). It follows that S ≤ O p (G) = R by Claim (b). Hence, S ≤ R ∩ V = W and so S G = S P O p (G) = S W ≤ W by [2, Lemma 1.2.16], which implies that S = 1. Then, T is a complement to V in G, so T is p-nilpotent. Now let V be any maximal subgroup of P containing R, and let M be a maximal subgroup of G such that G = R M. Then, M G/R is p-nilpotent, so M is a pnilpotent supplement to V in G. Thus, every maximal subgroup of P has a p-nilpotent supplement in G. Therefore, G is p-nilpotent by Lemma 2.7. This contradiction shows that we have (8).
The final contradiction. Claims (2) and (8) imply that E = P is a normal psubgroup of G. Let R be a minimal normal subgroup of G contained in P. Then, P/R is hypercyclically embedded in G by Claims (3) and (6). Therefore, R (P) by Lemma 2.8 and [20, III, Hilfsatz 3.3(a)]. Hence, (P) = 1, so P is an elementary abelian p-group. If |R| = p, then P is hypercyclically embedded in G by the Jordan-Hölder theorem for the chief series. Hence, R is not cyclic. Moreover, R is the unique minimal normal subgroup of G contained in P. Indeed, suppose that for some minimal normal subgroup N = R of G we also have N ≤ P. Then, P/N is hypercyclically embedded in G and so from the G-isomorphism R N /N R we get that |R| = p, a contradiction.
Let W be a maximal subgroup of N such that W is normal in a Sylow p-subgroup G p of G. Then, W = 1. We show that W is S-quasinormal in G. Let B be a complement to N in P and H = W B. Then, H is a maximal subgroup of P and W = H ∩ R. Therefore, W is S-quasinormal in G in the case when H is S-quasinormal in G by Lemma 2.1 (5). From now on, we suppose that H is not S-quasinormal in G.
Assume that H has a p-nilpotent supplement U in G and let S be the normal pcomplement in U . Then, P = P ∩ HU = H (P ∩ U ), where P ∩ U is normal in G since P is abelian. Moreover, 1 < P ∩ U < P since G is not p-nilpotent. Therefore, R ≤ P ∩ U . Then, [R, S] = 1, so G/C G (R) is a p-group and so C G (R) = G since R is a p-group. But then |R| = p. This contradiction shows that H has no p-nilpotent supplements in G and hence H is m-S-complemented in G by hypothesis.
Let S and T be subgroups of G such that S is generalized S -quasinormal in G and we have G = H T and H ∩ T ≤ S ≤ H . And let S = AB, where A is modular and B is S-quasinormal in G. Then, N H and so A G = 1, which implies that A G is hypercyclically embedded in G by Lemma 2.4. But then A = 1 since otherwise N ≤ A G ∩ P and so |N | = p. Therefore, S = B is S-quasinormal in G. Since T ∩ H ≤ S ≤ H and H is not S-quasinormal in G, it follows that T < G and for the normal subgroup T ∩ P of G we have 1 < T ∩ P. Then, N ≤ T and so N ∩ H = N ∩ S = W , which implies that W is S-quasinormal in G by Lemma 2.1 (5). Lemma 1.2.16] and so W = 1. Therefore, N is cyclic. This contradiction completes the proof of the result.
Proof of Theorem 1.4. Suppose that this theorem is false and consider a counterexample (G, E) for which |G| + |E| is minimal. Let p be the smallest prime dividing |E| and let P be a Sylow p-subgroup of E.
Then, E is p-supersoluble by Theorem 1.3 and so E is p-nilpotent since p is the smallest prime dividing |E| (see Claim (2) in the proof of Theorem 1.3). Note also that if X is a non-identity Hall subgroup of E, then X = E. Indeed, the hypothesis holds for (G/X , E/X ) and for (G, X ) by Lemma 2.3 (1). Hence in the case X = E, the choice of G implies that E/X and X are hypercyclically embedded in G. Hence, E is hypercyclically embedded in G by the Jordan-Hölder theorem for the chief series. This contradiction shows that E = P, so E is hypercyclically embedded in G by Theorem 1.3. The theorem is proved.
Proof of Theorem 1.5. This theorem is a corollary of Theorem 1.4 and Lemmas 2.5 and 2.6.

Some Applications of the Results
Theorems 1.3, 1.4 and 1.5 cover many known results. In particular, from Theorem 1.5, we get the following known results. Corollary 4.1 (Srinivasan [21]). If the maximal subgroups of the Sylow subgroups of G are S-quasinormal in G, then G is supersoluble.

Corollary 4.2 (Asaad [22]). Let F be a saturated formation containing all supersoluble groups and G a group with a normal subgroup E such that G/E ∈F . If G/E ∈F and every maximal subgroup of every Sylow subgroup of E is S-quasinormal in G, then G ∈F .
A subgroup H of G is said to be c-normal in G [23], if there is a normal subgroup T of G such that G = H T and H ∩ T ≤ H G . It is clear that every c-normal subgroup of G is also m-S-complemented in G. Hence, we get from Theorem 1.5 the following known results.       (Wei,Wang,Li [28]). Let F be a saturated formation containing all supersoluble groups and G a group with a normal subgroup E such that G/E ∈F . If all maximal subgroups of the Sylow subgroups of F * (E) are c -normal in G, then G ∈F. Corollary 4.9 (Asaad, Ramadan, Shaalan [29]). Let E be a soluble normal subgroup of G with supersoluble quotient G/E. Suppose that all maximal subgroups of any Sylow subgroup of F(E) are S-quasinormal in G. Then, G is supersoluble. Corollary 4.10 (Li, Wang [30]). Let F be a saturated formation containing all supersoluble groups and G a group with a normal subgroup E such that G/E ∈ F. If all maximal subgroups of any Sylow subgroup of F * (E) are S-quasinormal in G, then G ∈ F. Corollary 4.11 (Li,Wang [30]). Let F be a saturated formation containing all supersoluble groups and G a group with a normal subgroup E such that G/E ∈ F. If every maximal subgroup of every Sylow subgroup of F * (E) is S-quasinormal in G, then G ∈ F. Corollary 4.12 (Wei [28]). Let F be a saturated formation containing all supersoluble groups and G a group with a normal subgroup E such that G/E ∈ F. If every maximal subgroup of every Sylow subgroup of E is c-normal in G, then G ∈ F.
In view of Example 1.2(ii), we get also from Theorem 1.5 the following known results. Corollary 4.13 (Wei,Wang and Li [31]). Let F be a saturated formation containing all supersoluble groups and G a group with a normal subgroup E such that G/E ∈ F. If every maximal subgroup of every Sylow subgroup of F * (E) is c-supplemented in G, then G ∈ F. Corollary 4.14 (Ballester-Bolinches and Guo [32]). Let F be a saturated formation containing all supersoluble groups and G a group with a normal subgroup E such that G/E ∈ F. If every maximal subgroup of every Sylow subgroup of E is c-supplemented in G, then G ∈ F.