EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 11, No. 1, 2018, 160-168 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global The influence of C- Z-permutable subgroups on the structure of finite groups M. M. Al-Shomrani1,∗, A. A. Heliel2 1 Department of Mathematics, Faculty of Science, Northern Border University, Arar, Saudi Arabia 2 Department of Mathematics, Faculty of Science 62511, Beni-Suef University, Beni-Suef, Egypt Abstract. Let Z be a complete set of Sylow subgroups of a finite group G, that is, for each prime p dividing the order of G, Z contains exactly one and only one Sylow p-subgroup of G, say Gp. Let C be a nonempty subset of G. A subgroup H of G is said to be C-Z-permutable (conjugate- Z-permutable) subgroup of G if there exists some x ∈ C such that HxGp = GpH x, for all Gp ∈ Z. We investigate the structure of the finite group G under the assumption that certain subgroups of prime power orders of G are C-Z-permutable subgroups of G. 2010 Mathematics Subject Classifications: 20D10, 20D15, 20D20, 20F16. Key Words and Phrases: Sylow subgroup, Z-permutable subgroup, C-Z-permutable subgroup, p-nilpotent group, supersolvable group, Fitting subgroup, generalized Fitting subgroup, saturated formation. 1. Introduction Throughout this article only finite groups are considered. We use conventional no- tions and notation, as in Doerk and Hawkes [2]. In addtion, π(G) denotes the set of distinct primes dividing |G| and Gp is a Sylow p-subgroup of the group G for some prime p ∈ π(G). Two subgroups H and K of a group G are said to be permutable if HK = KH, that is, HK is a subgroup of G. Recall that a subgroup H of a group G is S-permutable (or S-quasinormal) in G if H permutes with every Sylow subgroup of G. This concept was introduced by Kegel [7] in 1962. Recently, in 2003, Asaad and Heliel [1] introduced the concept of Z-permutability which generalizes S-permutability as follows: Let Z be a complete set of Sylow subgroups of a group G. A subgroup H of G is said to be Z-permutable in G if H permutes with every ∗Corresponding author. Email addresses: malshomrani@hotmail.com (M. M. Al-Shomrani), heliel9@yahoo.com (A. A. Heliel) http://www.ejpam.com 160 c© 2018 EJPAM All rights reserved. M. M. Al-Shomrani, A. A. Heliel / Eur. J. Pure Appl. Math, 11 (1) (2018), 160-168 161 member in Z. More recently, in 2013, Heliel and Al-Gafri [4] generalized the concept of Z-permutability by introducing a new subgroup embedding property, namely, the conjugate-Z-permutability. Let C be a nonempty subset of a group G and Z be a complete set of Sylow subgroups of G. A subgroup H of G is said to be C-Z-permutable subgroup of G if there exists some x ∈ C such that HxGp = GpH x, for all Gp ∈ Z. Remark 1.2 and Examples 1.3 and 1.4 in [4] show that C-Z-permutability is a nontrivial generalization of Z-permutability. This article may be viewed as a continuation of Heliel and Al-Gafri [4]. In fact, we extend and improve the following theorem: Theorem 1.1. [[4], Theorem 3.11] Let F be a saturated formation containing the class of supersolvable groups U and let Z be a complete set of Sylow subgroups of a group G. Then the following two statements are equivalent: (a) G ∈ F. (b) There is a normal subgroup H in G and a solvable normal subgroup C of F ∗(H) such that G/H ∈ F, and the maximal subgroups of Gp ∩F ∗(H) are C-Z-permutable subgroups of G, for all Gp ∈ Z, where F ∗(H) is the generalized Fitting subgroup of H. More precisely, we prove the following theorem: Theorem 1.2. Let F be a saturated formation containing the class of supersolvable groups U. Let Z be a complete set of Sylow subgroups of a group G and let C be a solvable normal subgroup of G. Then the following two statements are equivalent: (a) G ∈ F. (b) There is a normal subgroup H in G such that G/H ∈ F and the maximal subgroups of Gp ∩ F ∗(H) are C-Z-permutable subgroups of G, for all Gp ∈ Z. Remark 1.3. Let S(F ∗(H)) denotes the solvable radical of F ∗(H), that is, S(F ∗(H)) is the unique largest solvable normal subgroup of F ∗(H). In Theorem 1.1, C is a solvable normal subgroup of F ∗(H). Therefore, C is contained in S(F ∗(H)). Since S(F ∗(H)) is characteristic in F ∗(H) and F ∗(H) is normal in G, we have that S(F ∗(H)) is normal in G. So, the maximal subgroups of Gp ∩ F ∗(H) are S(F ∗(H))-Z-permutable subgroups of G, for all Gp ∈ Z, where S(F ∗(H)) is a solvable normal subgroup of G. Thus, Theorem 1.1 can be seen as an immediate consequence of Theorem 1.2. 2. Basic definitions and preliminaries In this section, we list some definitions and known results from the literature that will be used in the sequel. M. M. Al-Shomrani, A. A. Heliel / Eur. J. Pure Appl. Math, 11 (1) (2018), 160-168 162 Let F be a saturated formation. Then the F-residual, denoted by GF, is the unique smallest normal subgroup of G such that G/GF ∈ F. Throughout, U denotes the class of supersolvable groups which is a saturated formations, see [[5], Satz 8.6, p. 713]. A normal subgroup N of a group G is an F-hypercentral subgroup of G provided N possesses a chain of subgroups 1 = N0 E N1 E ... E Ns = N such that Ni+1/Ni is an F-central chief factor of G, see [[2], p. 387]. The product of all F-hypercentral subgroups of G is again an F-hypercentral subgroup, denoted by ZF(G), and called the F-hypercenter of G, see [[2], IV, 6.8]. For the formation U, the U-hypercenter of a group G, denoted by ZU(G), is the product of all normal subgroups N of G such that each chief factor of G below N has prime order. For more details about saturated formations, see [[2], IV]. For any groupG, the generalized Fitting subgroup F ∗(G) is the unique maximal normal quasinilpotent subgroup of G. In fact, F ∗(G) is an important characteristic subgroup of G and it is a natural generalization of F (G). The basic properties of F ∗(G) can be found in [[6], X 13]. We define F ∗1 (G) = F ∗(G) and F ∗i (G)/F ∗i−1(G) = F ∗(G/F ∗i−1(G)) for i > 1. Since F ∗(G) 6= 1 when G 6= 1, there exists an integer n such that F ∗n(G) = G. Let Z be a complete set of Sylow subgroups of a group G and let N be a normal sub- group of G. We denote the following families of subgroups of G, G/N and N , respectively: ZN = {GpN : Gp ∈ Z}, ZN/N = {GpN/N : Gp ∈ Z}, Z ∩N = {Gp ∩N : Gp ∈ Z}. The following lemmas will be used in the sequel. Lemma 2.1. Let Z be a complete set of Sylow subgroups of a group G, C be a nonempty subset of G and N be a normal subgroup of G. Then: (a) Z∩N and ZN/N are complete sets of Sylow subgroups of N and G/N , respectively. (b) If U is C-Z-permutable subgroup of G, then UN/N is CN/N -ZN/N -permutable subgroup of G/N . (c) If U ≤ N, C ⊆ N and U is C-Z-permutable subgroup of G, then U is C-Z ∩ N - permutable subgroup of N . (d) Suppose that N ≤ U . Then U is C-Z-permutable subgroup of G if and only if U/N is CN/N -ZN/N -permutable subgroup of G/N . (e) If U is C-Z-permutable subgroup of G, then U ∩N is C-Z-permutable subgroup of G. Proof . For (a), see [[1], Lemma 2.1(a)]. For (b), (c) and (d); see [[4], Lemma 2.1]. (e) Let Gp be any member of Z. By hypothesis, there exists some x ∈ C such that UxGp is a subgroup of G, that is, UxGp = GpU x. Let K = Ux. It is clear that GpKN is a subgroup of G as N is normal in G. Since Gp ∩ KN is a p-subgroup of KN and M. M. Al-Shomrani, A. A. Heliel / Eur. J. Pure Appl. Math, 11 (1) (2018), 160-168 163 |KN : Gp ∩KN | = |GpKN : Gp| is a p′-number as Gp is a Sylow p-subgroup of GpKN , it follows that Gp∩KN is a Sylow p-subgroup of KN . Also, Gp∩K is a p-subgroup of K and |K : Gp ∩K| = |KGp : Gp| is a p′-number. Consequently, Gp∩K is a Sylow p-subgroup of K. Therefore, Gp∩K∩N = (Gp∩K)∩ (K∩N) is a Sylow p-subgroup of K∩N as K∩N is a normal subgroup of K. If M is a subgroup of G, let |M |p denotes the largest power of p dividing the order of M . As |KN | = |K||N | |K∩N | , then |KN |p = |K|p|N |p |K∩N |p . Clearly, (Gp ∩K )(Gp∩N) ≤ Gp as Gp∩N is a normal subgroup of Gp and so (Gp∩K)(Gp∩N) ≤ Gp∩KN . Now |(Gp ∩K)(Gp ∩N)| = |Gp∩K||Gp∩N | |Gp∩K∩N | = |K|p|N |p |K∩N |p = |KN |p = |Gp ∩ KN | and hence (Gp ∩ K)(Gp ∩ N) = Gp ∩ KN . By [[2], Lemma 1.2, p. 2], Gp(K ∩ N) = GpK ∩ GpN which is a subgroup of G as GpK and GpN are subgroups of G. So, we have x ∈ C such that Gp(U ∩N)x = Gp(U x ∩N) = Gp(K ∩N) is a subgroup of G, for all Gp ∈ Z. Thus, U ∩N is C-Z-permutable subgroup of G. Lemma 2.2. Let Z be a complete set of Sylow subgroups of a group G and C be a nonempty subset of G. Assume that H is a normal subgroup of G such that the maximal subgroups of Z ∩H are C-Z-permutable subgroups of G. Then for any nontrivial normal subgroup N of G, the maximal subgroups of (ZN/N) ∩ (HN/N) are CN/N -ZN/N -permutable subgroups of G/N . Proof . See [[4], Lemma 2.2]. Lemma 2.3. Let G be a group. Then: (a) F ∗(G) = F (G)E(G) and [F (G), E(G)] = 1, where E(G) is the layer subgroup of G. (b) F ∗(F ∗(G)) = F ∗(G) ≥ F (G); if F ∗(G) is solvable, then F ∗(G) = F (G). (c) CG(F ∗(G)) ≤ F (G). (d) Suppose that N is a normal subgroup of G contained in Φ(G), then F ∗(G/N) = F ∗(G)/N . Proof . (a), (b) and (c) can be found in [[6], X 13]. For (d), see [[10], Lemma 2.3 (8)]. Lemma 2.4. Let Z be a complete set of Sylow subgroups of a group G and C be a nonempty subset of G. Suppose that P is a normal p-subgroup of G, where p is a prime, and N is a minimal normal subgroup of G with N ≤ P . If N is complemented in P and the maximal subgroups of P are C-Z-permutable subgroups of G, then the order of N is p. Proof . By hypothesis, there exists a subgroup H of P such that P = NH and N ∩H = 1. Obviously, we have that N ≤ Gp ∈ Z. Let N/M be a chief factor of Gp. Then, the order of N/M is p. Clearly, MH is a subgroup of P as M is normal in P . Since M∩ H = M ∩ (N ∩ H) = 1, then |P : MH| = |NH : MH| = |N : M | = p and hence MH is a maximal subgroup of P . By hypothesis, MH is C-Z-permutable subgroup of G. Therefore, M = M(H ∩ N) = MH ∩ N is C-Z-permutable subgroup of G by Lemma M. M. Al-Shomrani, A. A. Heliel / Eur. J. Pure Appl. Math, 11 (1) (2018), 160-168 164 2.1(e). So, there exists some x ∈ C such that MxGq is a subgroup of G, for all Gq ∈ Z. This implies that MGx−1 q is a subgroup of G, for all Gq ∈ Z. Assume that q 6= p. Since M = M(N ∩ Gx−1 q ) = N ∩MGx−1 q and N ∩MGx−1 q is normal in MGx−1 q , it follows that Gx−1 q ≤ NG(M). If q = p, then M is normal in Gp and so Gp ≤ NG(M). Therefore, NG(M) = G and hence M is normal in G. But N is a minimal normal subgroup of G and M is a maximal subgroup of N , thus M = 1 and the order of N is p. Lemma 2.5. Let G be a group. Then: (a) E(G), the layer subgroup of G, is a perfect quasinilpotent characteristic subgroup of G. (b) If M is a perfect quasinilpotent subnormal subgroup of G, then M ≤ E(G). (c) If M is a solvable subgroup of G and E(G) ≤ NG(M), then [E(G),M ] = 1. Proof . For (a), see [[6], Definition 13.14, p. 128]. For (b) and (c), see [[6], Theorem 13.15(a), p. 128 and Lemma 13.16(b), p. 128–129], respectively. Lemma 2.6. Let Z be a complete set of Sylow subgroups of a group G and C be a solvable normal subgroup of G. If p is the smallest prime dividing the order of G and the maximal subgroups of Gp ∈ Z are C-Z-permutable subgroups of G, then G is p-nilpotent. Proof . See [[4], Theorem 3.1]. Lemma 2.7. Let F be a saturated formation containing the class of supersolvable groups U, Z be a complete set of Sylow subgroups of a group G and C be a solvable normal subgroup of G. Then the following two statements are equivalent: (a) G ∈ F. (b) There is a normal subgroup H in G such that G/H ∈ F and the maximal subgroups of Gp ∩H are C-Z-permutable subgroups of G, for all Gp ∈ Z. Proof . See [[4], Theorem 3.2]. Lemma 2.8. Suppose that G is a finite non-abelian simple group. Then there exists an odd prime r ∈ π (G) such that G has no Hall {2, r}-subgroup. Proof . See [[8], Lemma 2.6]. M. M. Al-Shomrani, A. A. Heliel / Eur. J. Pure Appl. Math, 11 (1) (2018), 160-168 165 3. Results First, we prove the following lemma: Lemma 3.1. Let Z be a complete set of Sylow subgroups of a group G and C be a nonempty subset of G. Suppose that P is a normal p-subgroup of G. If the maximal subgroups of P are C-Z-permutable subgroups of G, then P ≤ ZU(G). Proof . Assume that the result is false and let G be a counterexample of minimal order. If Φ(P ) 6= 1, then the maximal subgroups of P/Φ(P ) are CΦ(P )/Φ(P )-ZΦ(P )/Φ(P )- permutable subgroups of G/Φ(P ) by Lemma 2.1(d). Then, by the minimal choice of G, P/Φ(P ) ≤ ZU(G/Φ(P )). Hence, by [[11], Theorem 7.19, p. 39], P ≤ ZU(G), a contradiction. Thus, we may assume that, Φ(P ) = 1 and so P is elementary abelian p-group. Let N be a minimal normal subgroup of G contained in P . Since N ∩Φ(P ) = 1 as Φ(P ) = 1, it follows, by [[2], Theorem 9.2(f), p. 30], that N is complemented in P . The hypothesis and Lemma 2.4 imply that the order of N is p. If N = P , then P ≤ ZU(G) by the definition of ZU(G), a contradiction. So, we may assume that N 6= P . It is easy to see that Φ(P/N) = 1. Let M/N be a maximal subgroup of P/N . Then M is a maximal subgroup of P as |P : M | = |P/N : M/N | = p. By hypothesis and Lemma 2.1(b), M/N is CN/N -ZN/N -permutable subgroup of G/N . So, the maximal subgroups of P/N are CN/N -ZN/N -permutable subgroups of G/N . Therefore, P/N ≤ ZU(G/N) by the minimal choice of G. But ZU(G/N) = ZU(G)/N by [[11], Lemma 7.1(ii), p. 30], then P ≤ ZU(G), a contradiction completing the proof of the lemma. Now we can prove: Theorem 3.2. Let F be a saturated formation containing the class of supersolvable groups U, Z be a complete set of Sylow subgroups of a group G and C be a solvable normal subgroup of G. Then the following two statements are equivalent: (a) G ∈ F. (b) There is a normal subgroup H in G such that G/H ∈ F, F ∗(H) = F (H) and the maximal subgroups of the Sylow subgroups of F (H) are C-Z-permutable subgroups of G, for all Gp ∈ Z. Proof . We need only to prove (b)⇒ (a). Let P be any Sylow p-subgroup of F (H). Clearly, P is normal in G. The hypothesis and Lemma 3.1 imply that P ≤ ZU(G). Since this is true for any Sylow p-subgroup of F (G), we have that F (H) ≤ ZU(G). Note that [GU, F (H)] ≤ [GU, ZU(G)] = 1 by [[2], Theorem 6.10, p. 390] and hence GU ≤ CG(F (H)). Therefore, G/CG(F (H)) is an epimorphic image of G/GU ∈ U ⊆ F and so G/CG(F (H)) ∈ U ⊆ F. Consequently, G/CH(F (H)) = G/ (CG(F (H)) ∩H) ∈ F as G/CG(F (H)) ∈ F and G/H ∈ F. But CH(F (H)) ≤ F (H) holds by Lemma 2.3(c) and the fact that F ∗(H) = F (H), then M. M. Al-Shomrani, A. A. Heliel / Eur. J. Pure Appl. Math, 11 (1) (2018), 160-168 166 G/F (H) is an epimorphic image of G/CH(F (H)), thus G/F (H) ∈ F. Applying Lemma 2.7 yields G ∈ F. This comletes the proof of the theorem. The following lemma is a criterion for the solvability of finite groups: Lemma 3.3. Let G be a group. Then the following two statements are equivalent: (a) G is solvable. (b) G has a complete set Z of Sylow subgroups such that Gp ∈ Z is Z-permutable subgroup of G, where p is the smallest prime dividing the order of G. Proof . (a)⇒ (b). Since G is solvable, then G has a Sylow basis S by [9, Theorem 9.3.11, p. 229]. Let p be the smallest prime dividing the order of G and let Gp be the Sylow p-subgroup of G in S. By the definition of the Sylow basis S, we have that GpGq is a subgroup of G, for all Gq ∈ S, where q is a prime. Thus we can take Z = S and we have Gp is Z-permutable subgroup of G. (b)⇒ (a). Assume that the result is false and let G be a counterexample of minimal order. By Feit-Thompson Theorem [3], we may assume that p = 2. Since G2 ∈ Z is Z-permutable subgroup of G, it follows that G2Gq is a subgroup of G, for every odd prime q dividing the order of G, where Gq ∈ Z. Therefore, G is not simple by Lemma 2.8. Let N be a nontrivial proper normal subgroup of G. Clearly, G2 ∩N and G2N/N are Sylow 2-subgroups of N and G/N , respectively. By Lemma 2.1(e), G2 ∩ N is Z-permutable subgroup of G. Therefore, G2 ∩N is Z ∩N -permutable subgroup of N by Lemma 2.1(c). Also, G2N/N is ZN/N -permutable subgroup of G/N by Lemma 2.1(b). If 2 divides the order of N , then N is solvable by the minimal choice of G and if 2 does not divide the order of N , then N is solvable by Feit-Thompson Theorem [3]. The same argument holds for G/N , thus G/N is solvable. Since N and G/N are solvable, we have that G is solvable, a contradiction completing the proof of the lemma. Proof of Theorem 1.2. We need only to prove (b)⇒ (a) as (a)⇒ (b) is true withH = 1. Let E(G) be the layer subgroup of G. Since C is a solvable normal subgroup of G, it follows, by Lemma 2.5(c), that [E(G), C] = 1 and so C ≤ CG(E(G)). By Lemma 2.3(a), we have that F ∗(H) = F (H)E(H). Moreover, E(H) is a perfect quasinilpotent characteristic subgroup of H by Lemma 2.5(a). Now E(H) char H and H is normal in G, then E(H) is normal in G. Note that E(H) ≤ E(G) by Lemma 2.5(b), and hence C ≤ CG(E(H)). Now we will show that E(H) is solvable. By Feit-Thompson Theorem [3], we may assume that 2 divides the order of E(H). Clearly, Z ∩ E(H) is a complete set of Sylow subgroups of E(H) by Lemma 2.1(a). Let U be a maximal subgroup of G2∩F ∗(H), where G2 ∈ Z. The hypothesis and Lemma 2.1(e) imply that U ∩E(H) is C-Z-permutable subgroup of G. So, there exists some x ∈ C such that (U ∩E(H))xGq is a subgroup of G, for all Gq ∈ Z. But (U ∩ E(H))x = U ∩ E(H) as x ∈ C ≤ CG(E(H)) ≤ CG(U ∩ E(H)), then (U ∩ E(H))Gq M. M. Al-Shomrani, A. A. Heliel / Eur. J. Pure Appl. Math, 11 (1) (2018), 160-168 167 is a subgroup of G, for all Gq ∈ Z. Therefore, U ∩ E(H) is Z-permutable subgroup of G. By Lemma 2.1(c), U ∩ E(H) is Z ∩ E(H)-permutable subgroup of E(H). Suppose that G2 ∩ E(H) = G2 ∩ F ∗(H). By the hypothesis and the previous arguments, the maximal subgroups of G2 ∩ E(H) are Z ∩ E(H)-permutable subgroups of E(H). Consequently, E(H) is 2-nilpotent by Lemma 2.6, where C = 1 in this case. So, E(H) = (G2∩E(H))K, where K is a normal Hall 2′-subgroup of E(H). Because K is solvable by Feit-Thompson Theorem [3] and G/K ∼= G2 ∩ E(H) is solvable, it follows that E(H) is solvable. Thus, we may assume that G2 ∩ E(H) is a proper subgroup of G2 ∩ F ∗(H). Then we can choose U to be a maximal subgroup of G2 ∩F ∗(H) such that G2 ∩E(H) ≤ U . Therefore, G2 ∩E(H) = U ∩E(H) as G2 ∩E(H) is a Sylow 2-subgroup of E(H). Now we have that G2∩E(H) = U ∩E(H), as we proved in the beginning, is Z∩E(H)-permutable subgroup of E(H). This implies that E(H) is solvable by Lemma 3.3. So, in either case, E(H) is solvable. But E(H) is perfect, then E(H) = 1 and therefore F ∗(H) = F (H). Applying Theorem 3.2 yields G ∈ F. This completes the proof of the theorem. The next theorem is an improvement of Theorem 3.12 in [4]: Theorem 3.4. Let F be a saturated formation containing the class of supersolvable groups U, Z be a complete set of Sylow subgroups of a group G and C be a solvable normal subgroup of G. Then the following two statements are equivalent: (a) G ∈ F. (b) There is a normal subgroup H in G such that G/H ∈ F and the maximal subgroups of Gp∩F ∗n(H) are C-Z-permutable subgroups of G, for all Gp ∈ Z, for some positive integer n. Proof . We need only to prove (b)⇒ (a) as (a)⇒ (b) is true with H = 1. If n = 1, then G ∈ F by Theorem 1.2. So, we may assume that n > 1. Let K = F ∗n−1(H). It is clear that (G/K)/(H/K) ∼= G/H ∈ F. By Lemma 2.2, the maximal subgroups of (ZK/K) ∩ (F ∗n(H)/K) = (ZK/K) ∩ F ∗(H/K) are CK/K-ZK/K- permutable subgroups of G/K. Hence, G/K ∈ F by Theorem 1.2. Thus G/F ∗n(H) ∼= (G/K)/(F ∗n(H)/K) ∈ F and the maximal subgroups of Gp ∩ F ∗n(H) are C-Z-permutable subgroups of G, for all Gp ∈ Z. Applying Lemma 2.7 yields G ∈ F. This completes the proof of the theorem. Acknowledgements This paper was financially supported by The Deanship of Scientific Research, Northern Border University, under the project no. 434-071. The authors would like to thank The Deanship of Scientific Research for their financial support. 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