EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 17, No. 2, 2024, 810-818 ISSN 1307-5543 – ejpam.com Published by New York Business Global Finite Groups with Certain Weakly S-permutable Subgroups N. Alsowait1, A. A. Heliel2,3,∗, M. M. Al-Shomrani2, A. S. Allehyani2 1 Department of Mathematics, Faculty of Science, Northern Border University, Arar, Saudi Arabia 2 Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia 3 Department of Mathematics and Computer Science, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt Abstract. Let G be a finite group. A subgroup H of G is said to be weakly S-permutable in G if G has a subnormal subgroup T such that G = HT and T ∩H ≤ HsG, where HsG is the subgroup of H generated by all those subgroups of H which are S-permutable in G. In this paper, we prove the following: For a Sylow p-subgroup P of G (p > 2), suppose that P has a subgroup D such that 1 < |D| < |P | holds and all subgroups H of P with |H| = |D| are weakly S-permutable in G. Then, the commutator subgroup G ′ is p-nilpotent. We certainly believe that this result will improve and extend a current and classical theories in the literature. 2020 Mathematics Subject Classifications: 20D10, 20D10, 20D20. Key Words and Phrases: Sylow subgroups , S-permutable subgroups, weakly S-permutable subgroups, p-nilpotent groups 1. Introduction All groups considered in this paper will be finite. A subgroup H of a group G is said to be permutable in G if H permutes with every subgroup of G, that is, HK ⩽ G for all K ⩽ G. A subgroup H of G is called S-permutable in G provided H permutes with all Sylow subgroups of G, i.e., HP = PH for any Sylow subgroup P of G. This concept was proposed by Kegel in [8]. In 1996, Wang [10], defined the concept of c-normality as follows: A subgroup H of a group G is said to be c-normal in G if G has a normal subgroup K such that G = HK and H ∩ K ⩽ HG, where HG = CoreG(H) is the largest normal subgroup of G contained in H. As a generalization of c-normality, a subgroup H of G is said to be c-supplemented in G if there exists a subgroup K of G such that G = HK and ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v17i2.5120 Email addresses: Nawaf.Lazzam@nbu.edu.sa (N. Alsowait), aahsalem1@kau.edu.sa (A. A. Heliel), malshomrani@hotmail.com (M. M. Al-Shomrani), asaeedallehyani@stu.kau.edu.sa (A. S. Allehyani) https://www.ejpam.com 810 © 2024 EJPAM All rights reserved. A. A. Heliel et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 810-818 811 H ∩K ⩽ HG, where HG = CoreG(H) is the largest normal subgroup of G contained in H (see [1]). A number of scholars have studied influence of special types of subgroups behavior on the group structure. For instance, Gaschütz and Itö ([7], Satz 5.7, p. 436) proved that a group G is solvable if all its minimal subgroups are normal (a minimal subgroup is a subgroup of prime order). In [4], Heliel proved a group G is solvable if each subgroup of prime odd order of G is c-supplemented in G. In 2015, Hijazi [5] proved that if each Sylow subgroup P of G has a subgroup D such that 1 < |D| < |P | and all subgroups H of P with |H| = |D| are S-permutable ( or c-normal) in G, then G is solvable. It is remarkable to mention that the research on c -normal subgroups has formed a series, which is similar to the series of S-permutable subgroups, however the two series are independent of each other. In 2019, Hijazi and Charaf [6] continued the above mentioned studies and proved: Let P be a Sylow p-subgroup of a group G, where p is an odd prime, and suppose P has a subgroup D such that 1 < |D| < |P | and all subgroups H of P with |H| = |D| are S-permutable in G. Then G ′ is p-nilpotent. In [9], Skiba generalized both of the concepts S-permutability and c-normality as follows : A subgroup H of G is said to be weakly S-permutable in G if there is a subnormal subgroup T of G such that G = HT and H ∩ T ⩽ HsG, where HsG is the subgroup of H generated by all those subgroups of H which are S-permutable in G. Our main purpose here is to use this more general concept, weakly S-permutable, to take the above mentianed investigations further. More precisely, we prove: Main theorem. Let p be an odd prime and let P be a Sylow p-subgroup of G. Suppose that P has a subgroup D such that 1 < |D| < |P | and all subgroups H of P with |H| = |D| are weakly S-permutable in G. Then G ′ is p-nilpotent. 2. Preliminaries In this section, we state some known results from the literature which will be used in proving our results. Lemma 1. (See [6, Theorem 2]) Let P be a Sylow p-subgroup of a group G, where p is an odd prime. If each subgroup of P of order p is S-permutable in G, then G ′ is p-nilpotent. Lemma 2. (See [9, Theorem 1.4]) Let F be a saturated formation containing all supersolv- able groups and G a group with a normal subgroup E such that G/E ∈ F. Suppose that every non-cyclic Sylow subgroup P of E has a subgroup D such that 1 < |D| < |P | and all subgroups H of P with order |H| = |D| and with order 2|D| (if P is a non-abelian 2-group and |P : D| > 2) not having a supersolvable supplement in G are weakly S-permutable in G. Then G ∈ F Lemma 3. (See [2, Theorem 10.6 (a), p. 36]) Let G be a finite group: (i) CG(F (G))F (G)/F (G) contains no non-trivial solvable normal subgroup of G/F (G). In particuler, CG(F (G)) ≤ F (G) when G solvable. A. A. Heliel et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 810-818 812 (ii) If N is a minimal normal subgroup of G, then F (G) ≤ CG(N), furthermore, if N is abelian, then N ≤ Z(F (G). Lemma 4. (See [9, Theorem 2.20]) Let A be a p-group of automorphisms of the p-group P of odd order. Assume that every subgroup of P with prime order is A-invariant. Then A is cyclic. Lemma 5. (See [9, Lemma 2.10]) Let G be a group and H ⩽ K ⩽ G. Then: (i) If H is S-permutable in G, then H is weakly S-permutable in G. (ii) Suppose that H is normal in G. Then K/H is weakly S-permutable in G/H if and only if K is weakly S-permutable in G. (iii) If H is weakly S-permutable in G, then H is weakly S-permutable in K. (iv) Suppose that H is normal in G. Then the subgroup HE/H is weakly S-permutable in G/H for every weakly S-permutable subgroup E in G satisfying (|H|, |E|) = 1. (v) Suppose that H is a p-subgroup for some prime p and H is not S-permutable in G. Assume that H is weakly S-permutable in G. Then G has a normal subgroup M such that |G : M | = p and G = HM . Lemma 6. (See [11, Lemma 2.3, p. 214]) If G is solvable and Φ(G) = 1, then Fit(G) is the direct product of (Abelian) minimal normal subgroups of G. Lemma 7. (See [9, Theorem 2.11]) Let N be an elementary abelian normal subgroup of a group G. Assume that N has a subgroup D such that 1 < |D| < |N | and every subgroup H of N satisfying |H| = |D| is weakly S-permutable in G. Then some maximal subgroup of N is normal in G. Lemma 8. (See [3, Theorem 3.2, p. 228]) If Op′ (G) = 1, then CG(Op(G)) ⊆ Op(G). 3. Results We first prove the following theorem: Theorem 1. Let P be a Sylow p-subgroup of a group G, where p is an odd prime and suppose that each subgroup of P of order p is weakly S-permutable in G. Then G ′ is p-nilpotent. Proof. We prove the theorem by induction on |G|. If each subgroup of P of order p is S-permutable in G, then G ′ is p-nilpotent by Lemma 1. Thus we may assume that there exists a subgroup H of P of order p such that H is not S-permutable in G. By hypothesis, H is weakly S-permutable in G. So, there exists subnormal subgroup K of G such that G = HK and K ∩H ≤ HsG. However, HsG is the subgroup of H generated by all those subgroups of H which are S-permutable in G, and |H| = p, then HsG = 1 and so A. A. Heliel et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 810-818 813 K ∩H = 1. Clearly, K ◁G. By induction on |G|, K ′ is p-nilpotent. Hence, if Op ′ (G) ̸= 1, the group G/Op′ (G) satisfies the hypothesis of theorem and so G ′ /(G ′ ∩ Op′ (G)) is p- nilpotent, by induction on |G|, which implies that G ′ is p-nilpotent. Thus we may assume that Op′ (G) = 1. Now we have that K ′ charK ◁G which implies that K ′ ◁G and moreover K ′ is p-group as Op′ (G) = 1. ThenK has a normal Sylow p-subgroup, say P1, and so P1◁G (note that P1 is characteritstic in K and K◁G). Also, K possesses a p ′ -Hall subgroup, say K1. The subgroup K1 is Abelian since K/K ′ is Abelian with K ′ = P1 and K/P1 ∼= K1. It is clear that G is solvable. If P ◁ G, then G/P ∼= K1 and therefore, by Lemma 2, (taking E = P , F ∗(E) = F ∗(P ) = F (P ) since P is solvable, F (P ) = P from the definition, D = H with order p, 1 < p < pn, and all subgroups H of P are weakly S-permutable) G is supersolvable, in particular, G ′ is p-nilpotent. So we may assume that P is not normal in G. As Op′ (G) = 1, P1 is characteristic in G and P ⋪ G, we have F (G) = P1 and since G is solvable, we have, by Lemma 3 (1), that CG(F (G)) ≤ F (G) = P1. Clearly, K1 is a p ′ -group of automorophisms of F (G) = P1. Hence, if each subgroup of P1 is S-permutable in K, then K1 is cyclic, by Lemma 6, and so p is the largest prime dividing |G| (otherwise we have a contradiction). This means that P ◁ G, a contradiction. Thus P1 contains a subgroup L of order p such that L is not S-permutable in K and consequently L is not S-permutable in G. By hypothesis, L is weakly S-permutable in G. Hence, there exists a subgroup K∗ of G such that G = LK∗, L ∩ K∗ = 1 and K∗ ◁ G. As above P2 ◁ G, where P2 is a Sylow p-subgroup of K∗. But P1 ̸= P2 because L ≤ P1 and L ≨ P1, then P = P1P2 ◁ G, a contradiction completing the proof of the theorem. As a corollary of Theorem 1: Corollary 1. If each subgroup of prime order of G is weakly S-permutable in G, then G is solvable, L ◁ G ′ and G ′ /L is nilpotent, where L is a Sylow 2-subgroup of G ′ . Proof. By Theorem 1, G ′ is p-nilpotent for each odd prime p dividing |G|. So G ′ /L is nilpotent, L is a Sylow 2-subgroup of G ′ and hence G is solvable. Now, we are equipped to prove the Main Theorem: Proof. Assume that the result is false and let G be a counterexample of minimal order. Then: (1) Op′ (G) = 1. Assume that Op′ (G) ̸= 1, then, by Lemma 5 (4), G/Op′ (G) satisfies the hypothesis of the theorem. Hence (G/Op′ (G)) ′ = G ′ Op′ (G)/Op′ (G) ∼= G ′ /G ′ ∩ Op′ (G) is p-nilpotent by the minimal choice of G and so G ′ is p-nilpotent, a contradiction. (2) |D| > p. Assume that |D| = p. Then, by Theorem 1, G ′ is p-nilpotent, a contradiction. (3) There exists a subgroup H of P with |H| = |D| such that H is not S-permutable in G. Assume that all subgroups H of P with |H| = |D| are S-permutable in G. Then G ′ is p-nilpotent, by Lemma 1, a contradiction. A. A. Heliel et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 810-818 814 Now, We distinguish two cases: Case 1. |P : D| > p. Then, (4) G is solvable and F (G) is a maximal subgroup of P . By (3), There exists a subgroup H of P with |H| = |D| such that H is not S-permutable in G. Then, by the hypothesis, H is weakly S-permutable in G, that is, there exists a subnor- mal subgroup T of G such that G = HT and T ∩H ≤ HsG. Since H is not S-permutable in G, we have that HsG ̸= H and so T ̸= G. Then G has a normal subgroup M such that T ≤ M and |G/M | = p. Let A be a Sylow p-subgroup of M . Since |P : D| > p, we have that A has a subgroup D with 1 < D < A. Then, by the hypothesis, all subgroups L of A with |L| = |D| are weakly S-permutable in G and so all subgroups L of A with |L| = |D| are weakly S-permutable in M by Lemma 5 (iii). Then M ′ is p-nilpotent by choice of G. Hence, M ′ ≤ A as Op′ (G) = 1 by (1), and so A is characteristic in M and since M is normal in G, we have A ◁ G. Since M/A is Abelian and |G/M | = p, we have that G is solvable. Clearly, as A is a normal nilpotent subgroup of G, A ≤ F (G). Then, by (1), F (G) is a p-group. Since G is solvable, we have that G has a p ′ -Hall subgroup K and so K ≤ M (as G = HT and T ≤ M). Then K is Abelian. Hence if P = F (G), G/P ∼= K and so P ≥ G ′ , that is, G ′ is p-nilpotent, contradiction. Then A = F (G). (5) Φ(G) ̸= 1. Assume that Φ(G) = 1. Since G is solvable, from (4), it follows, by Lemma 6, that A = F (G) is a direct product of Abelian minimal normal subgroups of G. But A = F (G) has a maximal subgroup B such that B is normal in G. Then, by [3, A. (913), p. 33], for some minimal normal subgroup L of G contained in A = F (G), we have |L| = p. Then G/CG(L) is Abelian and G ′ ≤ CG(L). Since |D| > p from (2), then G/L satifies the hypothesis of the Lemma 5, and so (G/L) ′ ∼= G ′ /(G ′ ∩ L) is p-nilpotent by the choice of G and since G ′ ≤ CG(L), we have that G ′ is p-nilpotent, contradiction. (6) |Φ(G)| ≥ |D|. Assume that |Φ(G)| < |D|. By (4), Φ(G) < F (G) = A < P . Then G/Φ(G) satifies the hy- pothesis of the Lemma 5, so (G/Φ(G)) ′ = G ′ Φ(G)/Φ(G) ∼= G ′ /(G ′ ∩Φ(G)) is p-nilpotent by the choice of G and it follows easily that G ′ is p-nilpotent, contradiction. (7) Let L be a minimal normal subgroup of G such that L ≤ Φ(G). Then |L| ≤ |D|. Assume that |L| > |D|. Then every subgroup of L with order equals to |D| is weakly S-permutable in G and so, by Lemma 7, some maximal subgroup of L is normal in G. Then |L| = p > |D| which contradicts (2). Thus |L| ≤ |D|. (8) There exists a subgroup L1 of A = F (G) with |L1| = |D| such that is not S- permutable. Assume that all subgroups of L1 of A with |L1| = |D| are S-permutable. By (5), Φ(G) ̸= 1 and so Φ(G) contains a minimal normal subgroup L such that |L| ≤ |D| by (7). Consider A. A. Heliel et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 810-818 815 the factor group G/L. If |L| = |D|, then every subgroup of A/L of order p is S-permutable in G/L. Since L ≤ Φ(G), we have that F (G/L) = F (G)/L = A/L. Since G is solvable by (4), we have that C(G/Φ(G))(F (G/L)) = C(G/Φ(G))(F (G)/L) ≤ F (G/L) = F (G)/L = A/L. Then K̄ = KL/L ∼= K is a p ′ -group of automorphisms of A/L and every subgroup of A/L of prime order is K̄ invariant. Then K̄ ∼= K is cyclic by Lemma 6. Also, as G is solv- able, G contains a Hall subgroup PQ, where Q is a Sylow q-subgroup of G and q ̸= p. Hence, if p < q, PQ is p-nilpotent and so Q ≤ CG(A) = CG(F (G)) ≤ F (G) = A, a contradiction. Thus p is the largest prime dividing |G|. Since K is cyclic, we have by Burnside’s Theorem [5, Satz 2.8, p. 420], that P is normal in G, a contradiction. Thus that assume |L| < |D|. It is easy to see that G/L satisfies the hypothesis of the theorem and so (G/L) ′ ∼= G ′ /G ′ ∩ L is p-nilpotent by the choice of G and, since G ′ ∩ L ≤ Φ(G), we have that G ′ is p-nilpotent, a contradiction. (9) Finishing the proof of Case 1. By (8), there exists a subgroup L1 of A = F (G) with |L1| = |D| such that is not S- permutable. By the hypothesis, L1 is weakly S-permutable in G. Then there exists a sub- normal subgroup T1 of G such that G = L1T1 and T1 ∩L1 ≤ (L1)sG ̸= L1 and so T1 ̸= G. Hence, there exists normal subgroup M1 of G such that T1 ≤ M1 and |G/M1| = p. By Lemma 5, all subgroups L2 of P2, where P2 is a Sylow p-subgroup of M1 with |L2| = |D| are weakly S-permutable in M1. Then M ′ 1 is p-nilpotent by the choice of G. As Op′ = 1 from (1), we have M ′ 1 ≤ P2. Then P2 is characteristic in M1, and since M1 ⊴ G, it follows that P2 ⊴ G. Since G = L1T1 = L1M1, P2 ⊴ G, and L1 ⊴ A ⊴ M ⊴ G, we have that P = L1P2 is a subnormal Hall subgroup of G and so P ⊴ G, a contradiction. Case 2. |P : D| = p. Then, (10) There exists a maximal subgroup L of P with |L| = |D| such that L is not S- permutable in G. Assume that all maximal subgroups L of P with |L| = |D| are S-permutable in G. Then by Lemma 1, G ′ is p-nilpotent, a contradiction. (11) There exists a proper normal subgroup M of G such that |G/M | = p, G = LT , L ∩ T = (L)sG, where T is a subnormal subgroup of G and T ≤ M < G. By (10), L is not S-permutable in G. Then by the hypothesis, L is weakly S-permutable in G. Hence there exists a subnormal subgroup T in G such that G = LT and T ∩ L ≤ (L)sG ̸= L, since L is not weakly S-permutable in G. So T ̸= G and there exists a normal subgroup M of G such that |G/M | = p and T ≤ M < G. (12) (L)sG ̸= 1. Assume that (L)sG = 1. Then, by (11), G = LT and T ∩ L = 1, where T is a subnor- mal subgroup of G and there exists a normal subgroup M of G such that |G/M | = p and T ≤ M < G. Let P1 be a Sylow p-subgroup of M . If P1 is S-permutable in A. A. Heliel et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 810-818 816 G, then P1 ⊴ G. If Φ(P1) ̸= 1, then G/Φ(P1) satisfies the hypothesis of the theo- rem and hence (G/Φ(P1)) ′ ∼= G ′ /G ′ ∩ Φ(P1) is p-nilpotent by the choice of G, and since G ′ ∩ Φ(P1) ≤ Φ(G), we have that G ′ is p-nilpotent, a contradiction. Thus Φ(P1) = 1. Now it is clear that P1 ∩ T is a normal Sylow p-subgroup of T of order p. Then T/CT (P1 ∩ T ) is Abelian and T ′ ≤ CT (P1 ∩ T ) which implies that T ′ = P1 ∩ T . By Shur-Zassenhaus Theorem, T = (P1 ∩ T )K, where K is an abelian p ′ -Hall subgroup of T . Since G = L(P1 ∩ T )K = PK, that is, G is product of two nilpotent groups, then G is solvable by Kegel-Wielandt Theorem. Thus P1 = Op(G) = F (G). We can assume that P ⋬ G (otherwise, G ′ ≤ P , a contradiction). If Φ(G) ̸= 1, then Φ(G) < P1 = F (G). By choice of G, (G/Φ(G)) ′ = G ′ /G ′ ∩ Φ(G) is p-nilpotent and so G ′ is p-nilpotent, a contradiction. Thus Φ(G) = 1. Then P1 = F (G) a direct product of minimal normal subgroups of G. Hence, if L1 and L2 are two distinct minimal normal subgroups of G, then (G/L1) ′ and (G/L2) ′ are p-nilpotent by choice of G and so G ′ is p-nilpotent, a contradiction. Thus P1 = Op(G) = F (G) is the unique minimal normal subgroups of G by [2, A, 14.3], P1 ≤ NG(T ) which implies that T ⊴ M . Now as G is solvable, G contains a Hall subgroup PQ, where Q is a Sylow q-subgroup of G and p ̸= q. Hence, if p < q, PQ is p-nilpotent and so Q ≤ CG(F (G)) ≤ F (G), a contradiction. Thus p is the largest prime dividing |G|. Now T = (P1 ∩ T )K and K is not normal in T = (P1 ∩ T )K, otherwise K ⊴ T ⊴ M , that is, K ≤ Op′ (G) = 1, a contradiction. Hence T is a Frobenius group and so K is cyclic which implies that P ⊴ G, a contradiction. Thus P1 is not S-permutable in G. By the hypothesis, P1 is weakly S-permutable in G. Then there exists a subnormal subgroup K1 of G such that G = P1K1 and P1 ∩ K1 ≤ (P1)sG < P1. Hence, if (P1)sG = 1, then G = P1K1 and P1 ∩K1 = 1. Clearly, M = P1(M ∩K1) and M ∩K1 is subnormal in G. Since M ∩K1 is a p ′ -group, we have M ∩K1 ≤ Op′ (G) = 1 by (1), that is, M ∩K1 = 1 which means that P1 ⊴ G, a contradiction. Thus we may assume that (P1)sG ̸= 1. Then (P1)sG ≤ Op(G) ̸= 1. Assume that Φ(Op(G)) ̸= 1. Then G ′ /G ′ ∩ Φ(Op(G)) is p-nilpotent by the choice of G and so G ′ is p-nilpotent, a contradiction. Thus Φ(Op(G)) = 1 and so Op(G) = F (G) is elementary abelian. Assume that Φ(Op(G)) ≰ M . If Op(G) < |D|, then G ′ /G ′ ∩ Φ(Op(G)) = G ′ is p-nilpotent by the choice of G, a contradiction. Thus Op(G) = |D|. So Op(G) ∩M = 1 which means that |P | = p2 and Op(G) = |D| = p and this contradiction (2). Thus Op(G) ≤ M and Op(G) is a Sylow p-subgroup of M and Op(G) = P1, a contradiction as P1 is not S-permutable in G. Thus (L)sG ̸= 1. (13) Finishing the proof of Case 2. By (12), (L)sG ̸= 1. Then (L)sG ≤ Op(G) ̸= 1. Assume that Φ(Op(G)) ̸= 1. Then G ′ /G ′∩ Φ(Op(G)) is p-nilpotent by the choice of G, and so G ′ is p-nilpotent, a contradiction. Thus Φ(Op(G)) = 1, and so Op(G) is elementary abelian. If Op(G) < |D|, then G ′ /G ′ ∩Op(G) is p-nilpotent by the choice of G and so G is p-solvable. Since Op′ (G) = 1 by (1), we have, by Lemma 8, that CG(Op(G)) = Op(G). If Op(G) ∩ Φ(G) ̸= 1, then G ′ is p-nilpotent, a contradiction. Thus Op(G) is the unique minimal normal subgroups of G. Since M ⊴ G, we have Op(G) ∩ M = 1 or Op(G) ≤ M . If Op(G) ∩ M = 1, then G ′ is p-nilpotent, a contradiction. Assume that Op(G) ≤ M and let P1 be a Sylow p-subgroup of M . By hypothesis, P1 is weakly S-permutable in G. Then there exists a subnormal subgroup K1 REFERENCES 817 of G such that G = P1K1 and P1 ∩ K1 ≤ (P1)sG ≤ P1. If (P1)sG = P1, then P ⊴ G and this means that Op(G) = P1, a contradiction. Thus (P1)sG < P1. If (P1)sG = 1, then P1 ∩K1 = 1 which implies that Op(G) ∩K1 = 1 and so Op(G)K1 = Op(G)×K1, a contradiction. Thus we may assume that (P1)sG ̸= 1. Then (P1)sG ≤ Op(G). We agrue that Φ(G) = 1. If not, Op(G) ≤ Φ(G) which means that G ′ /G ′ ∩ Φ(G) is p-nilpotent and so G ′ is p-nilpotent, a contradiction. Thus Φ(G) = 1. Then there exists a maximal subgroup S of G such that G = Op(G)S, Op(G) ∩ S = 1. We agrue that Op(G) ≰ K1. If not, Op(G) ≤ K1. Then there exists a maximal subgroup V of P such that Op(G) ≰ V ( because if every a maximal subgroup V of P containing Op(G), then Op(G) ≤ Φ(P ) and so P = Op(G)(P ∩ S) = Φ(P )(P ∩ S) = Φ(P ) and this is impossible). This V is not S-permutable in G and so V is weakly S-permutable in G. Then there exists a subnormal subgroup T of G such that G = V T and V ∩ T ≤ (V )sG. Then (V )sG ≤ Op(G) and this implies that V ∩ T ≤ V ∩ Op(G) ≤ V ∩ T . Thus V ∩ Op(G) = V ∩ T and V ∩ T ≤ (V )sG ≤ V ∩Op(G). Now V ∩Op(G) = (V )sG is normal in P and (V )sG is S-permutable in G implies that (V )sG ⊴ G. Hence V ∩ T = (V )sG ≤ Op(G) ≤ V , a contradiction (note that (V )sG ̸= 1 because if (V )sG = 1, then Op(G) = p and G/CG(Op(G)) is abelian which means G ′ ≤ CG(Op(G)) and since G ′ /G ′ ∩ Op(G) is p-nilpotent, it follows that G ′ is p-nilpotent, a contradiction ). Thus Op(G) ≰ K1 and G/(K1)G is a p-group and since G ′ /G ′ ∩ Op(G) is p-nilpotent, a contradiction. Now we can assume that |Op(G)| = |D|. Then Op(G) is a maximal in P . Also Op(G) is elementary abelian and Op(G) ≰ M because P1 ∈ Sylp(M) is not S-permutable in G. Because Φ(G) is a p-group and Op(G) ≰ M , we have Φ(G) < Op(G) and Φ(G) = 1, that is, Op(G) is the unique minimal normal subgroups of G. Hence Op(G) ∩M = 1 and |Op(G)| = |D| = p, a contradiction with (2). As immediate consequences of the main theorem we have: Corollary 2. ([5], Theorem D) Suppose that each Sylow subgroup P of G has a subgroup D such that 1 < |D| < |P | and all subgroups H of P with |H| = |D| are c -normal in G. Then G is solvable. Corollary 3. ([6], Corollary 1) Let P be a Sylow p-subgroup of G (p > 2). Suppose that P has a subgroup D such that 1 < |D| < |P | and all subgroups H of P with |H| = |D| are permutable in G. Then G ′ is p-nilpotent. Acknowledgements The authors extend their appreciation to the Deanship of Scientific Research (DSR) at Northern Border University, Arar, KSA for funding this research work ”through the project number” NBU-FFR-2024-2089-01. Also, the authors thank the reviewers for their helpful suggestions and comments. References [1] A. Ballester-Bolinches, Y. Wang, and X. Guo. c-supplemented subgroups of finite groups. Glasgow Math. J., 42:383–389, 2000. REFERENCES 818 [2] K. Doerk and T. Hawkes. Finite soluble groups. Walter de Gruyter, 1992. [3] D. Gorenstein. Finite groups. AMS Chelsea Publishing, 1980. [4] A. A. Heliel. A note on c-supplemented subgroups of finite groups. Comm. Algebra, 42:1650–1656, 2014. [5] R. A. Hijazi. A note on solvability of finite groups. Journal of Advances in Mathe- matics, 10, 2015. [6] R. A. Hijazi and F. M. Charaf. Finite groups with certain permutability criteria. EJPAM, 12:571–576, 2019. [7] B. Huppert. Endliche gruppen i. Springer, Berlin-New York, 1979. [8] O. H. Kegel. Sylow-gruppen und subnormalteiler endlicher gruppen. Math. Z., 78:205– 221, 1962. [9] A. N. Skiba. On weakly s-permutable subgroups of finite groups. J. Algebra, 315:192– 209, 2007. [10] Y. Wang. c-normality of groups and its properties. J. Algebra, 180:954–965, 1996. [11] M. Weinstein. Between nilpotent and solvable. Polygonal Publishing House, Passaic, NJ, 1982.