On weakly Z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak{Z}}$$\end{document}-permutable subgroups of finite groups II

Let G be a finite group. We say that Z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak{Z}}$$\end{document} is a complete set of Sylow subgroups of G if for each prime p dividing the order of G,Z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${G, \mathfrak{Z}}$$\end{document} contains exactly one Sylow p-subgroup of G, Gp say. A subgroup of G is said to be Z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak{Z}}$$\end{document}-permutable in G if it permutes with every member of Z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak{Z}}$$\end{document}. A subgroup H of G is said to be weakly Z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak{Z}}$$\end{document}-permutable in G if there exists a subnormal subgroup K of G such that G = HK and H∩K≤HZ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${H \cap K \leq H_\mathfrak{Z}}$$\end{document}, where HZ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${H_{\mathfrak{Z}}}$$\end{document} is the subgroup of H generated by all those subgroups of H which are Z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak{Z}}$$\end{document}-permutable in G . In this paper, we prove that G is supersolvable if the maximal subgroups of Gp∩F*(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${G_{p} \cap F ^{\ast}(G)}$$\end{document} are weakly Z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak{Z}}$$\end{document}-permutable in G, for every Gp∈Z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${G_{p} \in \mathfrak{Z}}$$\end{document}, where F*(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${F^{\ast} (G)}$$\end{document} is the generalized Fitting subgroup of G. Also, 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 the class of all supersolvable groups, 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} if and only if there is a normal subgroup H in G such that G/H∈F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${G/H \in \mathfrak{F}}$$\end{document} and the maximal subgroups of Gp∩F*(H)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${G_{p} \cap F^{\ast}(H)}$$\end{document} are weakly Z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak{Z}}$$\end{document}-permutable in G, for every Gp∈Z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${G_{p} \in \mathfrak{Z}}$$\end{document}.

Recall that a subgroup H of a group G is said to be S-permutable in G if it permutes with every Sylow subgroup of G. This concept was introduced by Kegel [7], who called these subgroups S-quasinormal, and has been studied extensively by many authors. Asaad and Heliel [1] generalized S-permutability property by requiring permutability only with the members of a complete set of Sylow subgroups. We say that Z is a complete set of Sylow subgroups of G if for each prime p dividing |G|, Z contains exactly one Sylow p-subgroup of G. A subgroup H of G is said to be Z-permutable in G if H permutes with every member of Z. Following Wang [11], we say that a subgroup H of a group G is c-normal in G if there exists a normal subgroup K of G such that G = H K and H ∩ K ≤ H G , where H G = Core G (H ) is the largest normal subgroup of G contained in H . One can easily find groups with Z-permutable subgroups that are not c-normal and conversely there are also groups with c-normal subgroups that are not Z-permutable subgroups; see Examples 1, 2 and 3 in Heliel et al. [4]. In fact, there is no inclusion relationship between c-normality and Z-permutability. Consequently the authors in [4] unified and generalized both of Z-permutability and c-normality concepts as follows: a subgroup H of a group G is said to be weakly Z-permutable in G if there exists a subnormal subgroup K of G such that G = H K and H ∩ K ≤ H Z , where H Z is the subgroup of H generated by all those subgroups of H which are Z-permutable in G. Using this new subgroup embedding property, the authors in [4] studied the structure of a group G when all maximal subgroups of certain or every member of a complete set of Sylow subgroups of some normal subgroup of G are weakly Z-permutable in G. There they achieved results unified and generalized several recent results in the literature. The present paper may be viewed as a continuation of [4]; to be more precise, the following results have been proved in [4]: Recall that for any group G, the generalized Fitting subgroup F * (G) is the set of all elements x of G which induce an inner automorphism on every chief factor of G. F * (G) is an important characteristic subgroup of G and it is a natural generalization of the Fitting subgroup F(G). By [5, X 13], F * (G) = 1 if G = 1. The basic properties of F * (G) can be found in [5, X 13]. The reader is referred to [3] for basic properties and results about saturated formations.
(e) Suppose that P is a normal subgroup of G contained in O p (G) for some prime p, then F * (G/ (P)) = F * (G)/ (P).

is a solvable normal subgroup H in G such that G/H ∈ F and the maximal subgroups of the Sylow subgroups of F(H ) are weakly Z-permutable in G.
Proof See [4, Theorem 1.6].

Lemma 2.5 Let N be a normal subgroup of a group G and let Z be a complete set of Sylow subgroups of G.
Suppose that P is a p-subgroup of G for some prime p. Then: Proof Let Q be a Sylow q-subgroup of N . Since Q is characteristic in N and N G, we have Q G. Assume that q = p. As (|Q| , |P|) = 1 and the maximal subgroups of Q are weakly Z-permutable in G, we have, by Lemma 2.5(b), that the maximal subgroups of Q P/P are weakly ZP/P-permutable in G/P. Hence, we may assume that q = p and so P ≤ Q. Let M/P be a maximal subgroup of Q/P . By Lemma 2.5(a), M is a maximal subgroup of Q. The hypothesis and Lemma 2.3(b) imply that M/P is weakly ZP/P-permutable in G/P. Therefore, the maximal subgroups of Q/P are weakly ZP/P-permutable in G/P. Thus, the maximal subgroups of the Sylow subgroups of N /P are weakly ZP/P-permutable in G/P. Clearly, Z ∩ N is the set of all Sylow subgroups of N as N is nilpotent. Since ZP/P is a complete set of Sylow subgroups of G/P and N /P is a normal nilpotent subgroup of G/P, it follows that (ZP/P) ∩ (N /P) is the set of all Sylow subgroups of N /P. Thus, the maximal subgroups of (ZP/P) ∩ (N /P) are weakly ZP/P-permutable in G/P.

Proofs
Proof of Theorem 1.3. Assume that the result is false and let G be a counterexample of minimal order. Then the following statements about G are true: Since the maximal subgroups of G p are weakly Z-permutable in G, for all G p ∈ Z, it follows, by Lemma 2.1, that G is supersolvable, a contradiction. Thus, F * (G) = G.
(2) Every proper normal subgroup of G containing F * (G) is supersolvable. By (1), F * (G) = G and so there exists a proper normal subgroup N of G containing F * (G). Since Therefore, N is supersolvable by the minimal choice of G. Thus, every proper normal subgroup of G containing F * (G) is supersolvable.
(4) G has no normal subgroup of prime order and, therefore, Z (G) = 1. Let K be a normal subgroup of G of prime order. Assume that C G (K ) is a proper subgroup of G. By [3, p. 36, Theorem 10.6(b)] and (3), F * (G) = F(G) ≤ C G (K ). Therefore, C G (K ) is supersolvable by (2) and, since G/C G (K ) is isomorphic to a subgroup of the cyclic group Aut (K ), it follows that G is solvable. Thus, G is supersolvable by Lemma 2.4, a contradiction. Consequently we may assume that C G (K ) = G and so K ≤ Z (G). Lemma 2.2(f) and (3) imply that F * (G/K ) = F * (G)/K = F(G)/K . By Lemma 2.6, the maximal subgroups of the Sylow subgroups of F * (G/K ) = F(G)/K are weakly ZK /K -permutable in G/K . Therefore, G/K satisfies the hypothesis of the theorem and hence G/K is supersolvable by the minimal choice of G. This implies that G is supersolvable as K has a prime order, a contradiction. Thus, G has no normal subgroup of prime order and it follows easily that Z (G) = 1.
Let P be a Sylow p-subgroup of F(G), then the following holds: (5) (P) = 1 and, therefore, Pis abelian. Suppose that (P) = 1. Clearly, as (P) is characteristic in P and P G, (P) G. By Lemma 2.2(e) and (3), F * (G/ (P)) = F * (G)/ (P) = F(G)/ (P). Lemma 2.6 implies that the maximal subgroups of the Sylow subgroups of F * (G/ (P)) = F(G)/ (P) are weakly Z (P)/ (P)-permutable in G/ (P). Therefore, G/ (P) is supersolvable by the minimal choice of G. Since (P) ≤ (G), it follows that G/ (G) is supersolvable. But since the class of supersolvable groups is a saturated formation, then G is supersolvable, a contradiction. Thus, (P) = 1 and it follows that P is abelian. (3), it follows that P O p (G) is supersolvable by (2) and hence O p (G) is supersolvable. Therefore, G is solvable as G/O p (G) is a p-group. By Lemma 2.4, G is supersolvable, a contradiction. Thus, G = P O p (G).
is a normal subgroup of G for any maximal subgroup U of P.
Let x be an element of G and let V = U x −1 . It is clear that V is a maximal subgroup of P as P is normal in G and |V | = |U |. By hypothesis, V is weakly Z-permutable in G. Then there exists a subnormal subgroup K of G such that G = V K and V ∩ K ≤ V Z . Since P is abelian by (5) and G = P K , it follows that P ∩ K is a normal subgroup of G. Clearly, O p (G) ≤ K as K is subnormal in G and |G : K | is a power of p. By (6) q is a subgroup of G, for all x ∈ G and for all G q ∈ Z. This implies that x is a subgroup of G, for all x ∈ G and for all G q ∈ Z. As the Sylow subgroups of G are conjugate, we have that U ∩ O p (G) is an S-permutable in G. By [2, p. 17, Lemma 1.2.16], O p (G) ≤ N G (U ∩ O p (G)). Also, U ∩ O p (G) is a normal subgroup of P as P is abelian by (5).
. Also, P ≤ C G (P) as P is abelian by (5). Therefore, G = P O p (G) ≤ C G (P) by (6) and so P ≤ Z (G) which is a contradiction with (4). Thus, P ∩ O p (G) = 1.
(9) P ∩ (G) = 1. Suppose that P ∩ (G) = 1. By Lemma 2.9, P = L 1 × L 2 × · · · × L t , where each L i is a minimal normal subgroup of G, for i = 1, 2, . . . , t. Since P ∩ O p (G) = 1 by (8) and (P) = 1 by (5), then there exists a maximal subgroup U of P such that P = (P ∩ O p (G))U . This implies that |(P ∩ O p (G)) : is a chief factor of G with order p. Consider P as an operator group with operator domain = I nn(G). Then the series 1 L 1 L 1 L 2 · · · L 1 L 2 · · · L t = P is an -composition series of the -group P. Since is an -composition factor of P, it follows, by the Jordan-Hölder Theorem, that  (2) and so S is solvable. Since S/L is a solvable perfect group by Lemma 2.2(b), then S/L = 1 and it follows that F * (G/L) = F(G)/L. By Lemma 2.6, the maximal subgroups of the Sylow subgroups of F * (G/L) = F(G)/L are weakly ZL/L-permutable in G/L. Thus, G/L satisfies the hypothesis of the theorem and hence G/L is supersolvable by the minimal choice of G. Since L ≤ (G), it follows that G/ (G) is supersolvable. This implies that G is supersolvable as the class of supersolvable groups is a saturated formation, a contradiction. Thus, L is the unique minimal normal subgroup of G contained in P.
(12) The final contradiction. By (5), there exists a maximal subgroup U of P such that P = LU . If L ≤ U ∩ O p (G), then L ≤ U and hence P = U , a contradiction. Therefore, L is not contained in U ∩ O p (G). Since U ∩ O p (G) is a normal subgroup of G by (7) and L U ∩ O p (G), it follows, by (11), that U ∩ O p (G) = 1 . Clearly, (8)