EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 2, Article Number 5993 ISSN 1307-5543 – ejpam.com Published by New York Business Global 1 Finite Groups with Certain SSH-subgroups2 A. S. Allehyani3 Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah,4 Saudi Arabia5 6 Abstract. Let G be a finite group. A subgroup H of G is S-permutable in G if H permutes with every Sylow subgroup of G. A subgroup H of G is called an SSH-subgroup in G if G has an S-permutable subgroup K such that HSG = HK and Hg∩NK(H) ⩽ H, for all g ∈ G, where HSG is the intersection of all S-permutable subgroups of G containing H. In this paper, we investigate the structure of a finite group G under the assumption that certain subgroups of prime power orders are SSH-subgroups of G. 2020 Mathematics Subject Classifications: 20D10, 20D207 Key Words and Phrases: S-permutable subgroups, c-normal subgroups, H-subgroups, HC-8 subgroups, SSH-subgroups, supersolvable groups, saturated formations9 10 1. Introduction11 Throughout this paper, we assume that all groups in this paper are finite and G always12 denotes a group. Recall that a subgroup H of G is called permutable in G if H permutes13 with every subgroup of G, that is, HK ⩽ G, for all K ⩽ G; and a subgroup H is said14 to be S-permutable in G if H permutes with every Sylow subgroup of G. The concept of15 S-permutability as generalization of normality and permutability was defined by Kegel [1]16 in 1962.17 18 Another generalization of normality was given by Wang [2] in 1996 as follows: A19 subgroup H of G is said to be c-normal in G if G has a normal subgroup K such that20 G = HK and H ∩ K ⩽ HG, where HG = CoreG(H) = ∩g∈GH g is the largest normal21 subgroup of G contained in H. In 2000, the concept of H-subgroup was introduced by22 Bianchi et al. in [3] as follows: A subgroup H of G is called an H-subgroup in G if23 Hg ∩NG(H) ⩽ H, for all g ∈ G.24 Wei and Guo [4], in 2012, defined a new concept, named HC-subgroup, which is a25 generalization of c-normality and H-subgroup as follows: A subgroup H of G is said to26 be an HC-subgroup of G if there exists a normal subgroup K of G such that G = HK27 DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.5993 Email address: asaeedallehyani@stu.kau.edu.sa (A. S. Allehyani) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) A. S. Allehyani / Eur. J. Pure Appl. Math, 18 (2) (2025), 5993 2 of 13 and Hg ∩ NK(H) ≤ H, for all g ∈ G. Clearly, every c-normal subgroup of G is an HC-28 subgroup of G; to see that, if H is a c-normal subgroup of G, then there exists a normal29 subgroup K of G such that G = HK and H ∩ K ≤ CoreG(H). Thus Hg ∩ NK(H) =30 (H ∩K)g ∩NG(H) ≤ H, for all g ∈ G and so H is an HC-subgroup of G. However, the31 converse is not true in general (see [4, Example 1]). Moreover, it is easy to see that every32 H-subgroup of G is an HC-subgroup of G, but the converse is not true in general (see [4,33 Example 2]).34 35 In 2016, Asaad and Ramadan [5] introduced the concept of weakly HC-embedded sub-36 group as a generalization of HC-subgroup as follows: A subgroup H of G is said to be37 weakly HC-embedded in G if there exists a normal subgroup K of G such that HG = HK38 and Hg ∩ NK(H) ≤ H, for all g ∈ G, where HG = ∩{N : N ⊴ G and H ≤ N} is the39 normal closure of H in G.40 41 In 2018, AL-Gafri and Nauman [6] introduced the concept of SSH-subgroup which42 is a generalization of weakly HC-embedded subgroup as follows: A subgroup H of G43 is said to be an SSH-subgroup in G if G has an S-permutable subgroup K such that44 HSG = HK and Hg ∩ NK(H) ⩽ H, for all g ∈ G, where HSG is the intersection45 of all S-permutable subgroups of G containing H, that is, HSG = ∩{L ≤ G : H ⩽46 L and L is an S-permutable subgroup in G}. Clearly, every weakly HC-embedded in G47 is an SSH-subgroup in G; to see that, assume that H is weakly HC-embedded in G.48 Then there exists a normal subgroup T of G such that HG = HT and Hg ∩NT (H) ⩽ H,49 for all g ∈ G. Note that HSG is S-permutable in G and HSG ⩽ HG by Lemma 6. So,50 HSG = HSG ∩ HT = H ( HSG ∩ T ) = HK, where K = HSG ∩ T . Moreover, K is S-51 permutable in G by [1, Satz 2]. Clearly, Hg ∩ NK(H) = Hg ∩ NG(H) ∩ T ∩ HSG =52 Hg ∩NT (H) ∩HSG ⩽ H ∩HSG = H, for all g ∈ G. Thus H is an SSH-subgroup in G.53 But the converse is not true in general (see [6, Example 1.5]).54 55 Several researchers have studied the structure of finite groups by using the above men-56 tioned concepts. For example, in 1980, Srinivasan [7] proved that if all maximal subgroups57 of every Sylow subgroup of a group G are normal in G, then G is supersolvable. Wang58 [2] got the supersolvability of the group G when all maximal subgroups of every Sylow59 subgroup of G are c-normal in G. Moreover, Asaad in [8] proved that if all maximal sub-60 groups of every Sylow subgroup of G are H-subgroups in G, then G is supersolvable. In61 addition, in [4], Wei and Guo obtained the same previous result by replacing H-subgroup62 with HC-subgroup. Asaad and Ramadan [5], studied extensively the structure of a finite63 group by using the weakly HC-embedded subgroup concept and proved that: Let G be64 a group and P a Sylow p-subgroup of G. Then G is p-nilpotent if and only if NG(P ) is65 p-nilpotent and every maximal subgroup of P is weakly HC-embedded in G. In the same66 line of these studies, AL-Gafri and Nauman [6] used the SSH-subgroup concept to get a67 new structure of the group G. In fact, they proved that let P be a Sylow p-subgroup of a68 group G, for some prime p. Then G is p-nilpotent if and only if NG(P ) is p-nilpotent and69 every maximal subgroup of P is an SSH-subgroup in G. Also, they proved that a group70 A. S. Allehyani / Eur. J. Pure Appl. Math, 18 (2) (2025), 5993 3 of 13 G is supersolvable if and only if the maximal subgroups of the non-cyclic Sylow subgroups71 of G′ are SSH-subgroup in G. For more results along these same lines; see [9–12].72 73 The main aim of this paper is to continue the above mentioned investigations. More74 precisely, we study the structure of a finite group G under the assumption that certain75 subgroups of prime power orders are SSH-subgroups in G itself.76 77 Recall that a class of group F is said to be a formation if F is closed under taking78 epimorphic images and every groupG has a unique smallest normal subgroup with quotient79 in F. A formation F is called saturated if it is closed under taking Frattini extensions. U80 denotes the class of all supersolvable groups. Clearly, U is a saturated formation (see [13,81 p. 713, Satz 8.6]).82 Most of the notation is standard and can be found in [14] and [15]. In particular, |G|83 denotes the order of G. Moreover, Φ(G), F (G) and F ⋆(G) denote the Frattini subgroup,84 the Fitting subgroup and the generalized Fitting subgroup of G.85 2. Preliminaries86 In this section, we state some known results from the literature which will be used in87 proving our results.88 Lemma 1. Let H and L be normal subgroups of G and let p ∈ π(G). Then, the following89 hold:90 (i) Φ(H) ⩽ Φ(G).91 (ii) If L ⩽ Φ(G), then F (G/L) = F (G)/L.92 (iii) If L ⩽ H ∩ Φ(G), then F (H/L) = F (H)/L.93 Proof. For (i), see [13, III, Hilfssatz 3.3]. For (ii), and (iii), see [16, Lemma 2.7].94 Lemma 2. Let H,M and L be subgroups of a group G such that H is an SSH- subgroup95 in G and L ◁ G. Then the following statements hold:96 (i) If H ⩽ M , then H is an SSH-subgroup in M .97 (ii) Assume that L ⩽ M . Then M is an SSH-subgroup in G if and only if M/L is an98 SSH-subgroup in G/L.99 (iii) Assume that H is a p-subgroup of G and L is a p′-subgroup of G, for some prime p.100 Then HL and HL/L are SSH-subgroups in G and G/L, respectively.101 Proof. See [6, Lemma 2.4].102 A. S. Allehyani / Eur. J. Pure Appl. Math, 18 (2) (2025), 5993 4 of 13 Lemma 3. Let G be a group and let N be a nontrivial normal subgroup of G. If N∩Φ(G) =103 1, then F (N), the Fitting subgroup of N , is the direct product of the minimal normal104 subgroups of G which are contained in F (N).105 Proof. See [17, Lemma 2.6].106 Lemma 4. Let G be a group and let H be an H-subgroup in G. If H is subnormal in G,107 then H is normal in G.108 Proof. See [3, Theorem 6.2].109 Lemma 5. Let G be a solvable group. Suppose that F (G) possesses a normal series110 Φ(G) = K0 ⩽ Ki ⩽ K2 ⩽ . . . ⩽ Kn = F (G),111 such that Ki ,s are normal subgroups of G and |Ki/Ki−1| = prime (1 ≤ i ≤ n). Then G is112 supersolvable.113 Proof. See [13, p. 720, Satz 9.9].114 Lemma 6. Let G be a group and H ⩽ K ⩽ G. Then HSG is S-permutable in G and115 HSG ⩽ HG.116 Proof. See [18, Lemma 2.5(1)].117 Lemma 7. Let P be an elementary abelian p-subgroup of G such that P is not cyclic.118 Then the following statements are equivalent:119 (i) The subgroups of order p in P are normal in G.120 (ii) The maximal subgroups of P are normal in G.121 Proof. See [19, Lemma 2.6].122 Lemma 8. Let G be a group and let L be a subgroup of G:123 (i) If L ⊴ G, then F ∗(L) ⩽ F ∗(G).124 (ii) F ∗(F ∗(G)) = F ∗(G) ≥ F (G). If F ∗(G) is solvable, then F ∗(G) = F (G)125 (iii) Suppose that P is a normal p-subgroup, then F ∗(G/Φ(P )) = F ∗(G)/Φ(P )126 Proof. See [20, p. 123, X. 13].127 Lemma 9. Let G be a group and let H be a normal cyclic subgroup with G/H supersolv-128 able. Then G is supersolvable.129 Proof. See [21, Theorm 1.2].130 A. S. Allehyani / Eur. J. Pure Appl. Math, 18 (2) (2025), 5993 5 of 13 3. Main results131 In the present section, we will prove some theorems, also we will give some illustrative132 examples and counterexamples.133 134 We will begin our study with the following theorem:135 Theorem 1. Assume that G is a solvable group and all maximal subgroups of the non-136 cyclic Sylow subgroups of F (G) are SSH-subgroups of G. Then G is supersolvable.137 Proof. Assume that the result is false and let G be a counterexample of minimal order.138 We distinguish the following two cases:139 140 Case 1: Φ(G) ̸= 1.141 142 Then there exists a prime p such that p||Φ(G)|. Since Φ(G) ⩽ F (G), it follows that143 p||F (G)|. Let P1 be a non-cyclic Sylow p-subgroup of Φ(G). Since P1 is characteristic in144 Φ(G) ⊴ G, we have that P1 ⊴ G. By Lemma 1(ii), we have that F (G/P1) = F (G)/P1.145 Let P2/P1 be a maximal subgroup of the non-cyclic Sylow p-subgroup of F (G)/P1. Then146 P2/P1 is an SSH-subgroup in G/P1. By Lemma 2(ii), P2 is an SSH-subgroup in G.147 Also, if Q is the non-cyclic Sylow q-subgroup of F (G)/P1, then Q = FqP1/P1, where148 Fq is the non-cyclic Sylow q-subgroup of F (G)(q ̸= p). Let M/P1 be a maximal subgroup149 of FqP1/P1. Then M = (M ∩ Fq)P1, where M ∩ Fq is a maximal subgroup of Fq. By150 hypothesis of the theorem, M ∩ Fq an SSH-subgroup in G, which implies that M/P1 is151 an SSH-subgroup in G/P1 by Lemma 2(iii). Therefore, every maximal subgroups of the152 non-cyclic Sylow subgroup of F (G)/P1 are SSH-subgroups in G/P1. Then, by minimal-153 ity choice of |G|, G/P1 is supersolvable. Since (G/P1)/(Φ(G)/P1) ∼= G/Φ(G), we have154 G/Φ(G) is supersolvable. By a well-known Theorem of Huppert [13, p. 713, Satz 8.6], G155 is supersolvable, a contradiction.156 157 Case 2: Φ(G) = 1.158 159 Let P be a non-cyclic Sylow p-subgroup of F (G). Since P is characteristic in F (G) ⊴ G,160 it follows that P ⊴ G. By Lemma 1(i), Φ(P ) ⩽ Φ(G) and since Φ(G) = 1, then Φ(P ) = 1,161 for every Sylow subgroup of F (G). Since G is solvable and Φ(G) = 1, then by [13, p. 279,162 Staz 4.5], we have F (G) = R1 ×R2 ×R3 × · · · ×Rm, where Ri(i = 1, . . . ,m) is a minimal163 normal subgroup of G. Clearly, R1 ≤ P . If R1 = P , then P is a minimal normal subgroup164 of G. If |R1| = pe, e > 1, let P1 be a maximal subgroup of R1 = P . By hypothesis, P1 is165 an SSH-subgroup of G. Then G has an S-permutable subgroup T such that PSG = P1T166 and P g 1 ∩NT (P1) ≤ P1, for all g ∈ G. Since P is a minimal S-permutable subgroup of G,167 we have that PSG = P = P1T and P g 1 ∩NT (P1) ≤ P1, for all g ∈ G. Since P1 < P , we see168 that T ̸= 1 and so P = T as P is abelian and so P g 1 ∩NT (P1) = P g 1 ≤ P1, for all g ∈ G.169 This means that P1 is normal in G, a contradiction. Thus we may assume that R1 is a170 proper subgroup of P , where P is a non-cyclic Sylow p-subgroup of F (G). Since Φ(P ) = 1,171 A. S. Allehyani / Eur. J. Pure Appl. Math, 18 (2) (2025), 5993 6 of 13 then there exists a maximal subgroup P1 of P such that R1≮ P1. By hypothesis, P1 is172 an SSH-subgroup in G, then there exists an S-permutable subgroup K of G such that173 (P1) SG = P1K and (P1) g ∩NK(P1) ⩽ P1, for all g ∈ G. Assuming that K = P , then we174 have (P1) g ∩NG(P1) = (P1) g ∩K ∩NG(P1) = (P1) g ∩NK(P1) ⩽ P1. Then we get P1 is an175 H-subgroup in G and P1 ⊴ P . By Lemma 4, we get P1 ⊴ G. Now, P1 ∩R1 ⊴ G and R1 is176 a minimal normal subgroup of G, means that P1 ∩R1 = 1 and since P = P1R1 we have:177 p = |P : P1| = |R1 : P1 ∩R1| = |R1|.178 Thus, R1 is a cyclic subgroup of prime order.179 Set Ki = R1 ×R2 ×R3 × · · · ×Ri, where i = 1, . . . ,m and consider the chain180 1 = Φ(G) = K0 ⩽ K1 ⩽ K2 ⩽ . . . ⩽ Km = F (G).181 Clearly, Ki are normal subgroups of G and |Ki/Ki−1| = prime (1 ≤ i ≤ m). Applying182 Lemma 5, G is supersolvable, a contradiction.183 The following example shows that the solvability of G in Theorem 1 can not be omitted.184 Example 1. Consider the group G = N×M , where N is nilpotent and M is a non-abelian185 simple group. Clearly, G is not solvable. We notice that F (G) = F (N) = N and every186 maximal subgroup of the non-cyclic Sylow subgroups of F (G) is SSH-subgroup of G.187 The converse of Theorem 1 is not necessary true as the following example188 Example 2. Let G = Z/3Z × S3. Then G is supersolvable, but there exists a maximal189 subgroup of the non-cyclic Sylow subgroups of F (G) which is not SSH-subgroup of G.190 Theorem 2. Let G be a group with a normal solvable subgroup H such that G/H is191 supersolvable. If all maximal subgroups of the non-cyclic Sylow subgroup of F (H) are192 SSH-subgroups of G, then G is supersolvable.193 Proof. Assume that the claim is false and choose G to be a counterexample of minimal194 order. We distinguish the following two cases:195 196 Case 1: Φ(G) ∩H ̸= 1.197 198 Then there exists a prime p such that p||Φ(G) ∩ H|. Let P1 be a non-cyclic Sylow199 p-subgroup of (Φ(G) ∩ H). Since (Φ(G) ∩ H) is a nilpotent, then P1 ⊴ (Φ(G) ∩ H).200 Now, P1 is a normal Hall subgroup of (Φ(G) ∩ H) implies that P1 is characteristic in201 (Φ(G) ∩H) ⊴ G, hence P1 ⊴ G. So, (G/P1)/(H/P1) ∼= G/H is supersolvable.202 Now, we show that F (H/P1) = F (H)/P1. It is clear F (H)/P1 is a normal nilpo-203 tent subgroup of H/P1 and F (H/P1) is a largest normal nilpotent subgroup of H/P1 so,204 F (H)/P1 ⩽ F (H/P1). Set F (H/P1) = L/P1. Since L/P1 is characteristic in H/P1 ⊴205 G/P1, then L/P1 ⊴ G/P1. But L is a normal nilpotent subgroup of H holds by the206 fact that P1 ⩽ Φ(G), then L ⩽ F (H), and so L/P1 = F (H/P1) ⩽ F (H)/P1. There-207 fore, F (H)/P1 = F (H/P1). Let P2/P1 be a maximal subgroup of the non-cyclic Sylow208 A. S. Allehyani / Eur. J. Pure Appl. Math, 18 (2) (2025), 5993 7 of 13 p-subgroup of F (H)/P1. Then, by hypothesis, P2/P1 is an SSH-subgroup in G/P1. Thus,209 by minimalty choice of |G|, G/P1 is supersolvable and since P1 ⩽ Φ(G), we get G/Φ(G)210 is supersolvable. By Huppert’s Theorem [13, p. 713, Satz 8.6], G is supersolvable, a211 contradiction.212 213 Case 2: Φ(G) ∩H = 1214 215 IfH = 1, nothing need to prove. So, assume thatH ̸= 1. By Lemma 3, F (H) is a direct216 product of minimal normal subgroups of G which are contained in F (H). Let P be a non-217 cyclic Sylow p-subgroup of F (H). Then P = R1×R2×R3×· · ·×Rt, where Ri(i = 1, . . . , t)218 is a minimal normal subgroup of G. Then there exists a maximal subgroup P1 of P , and219 by hypothesis, P1 is an SSH-subgroup in G. Then there exists an S-permutable subgroup220 K of G such that (P1) SG = P1K and (P1) g ∩NK(P1) ⩽ P1, for all g ∈ G. Assuming that221 K = P , then we have (P1) g∩NG(P1) = (P1) g∩K∩NG(P1) = (P1) g∩NK(P1) ⩽ P1. Then222 we get P1 is an H-subgroup in G and P1 ⊴ P . Applying Lemma 4, we get P1 ⊴ G. Let223 Q be a non-cyclic Sylow q-subgroup of F (H) such that (p, |Q|) = 1. Now, P1Q ⩽ G and224 since P1 is a normal Hall subgroup of P1Q, it follows that P1 is a characteristic subgroup225 of P1Q. In particular, P1 is a normal subgroup of P1Q. Hence Q ⩽ NG(P1) for all Sylow226 q-subgroup Q of F (H), where (p, |Q|) = 1. Since P1 is a normal subgroup of P and P1 is a227 normal subgroup of P1Q, we get P1 is a normal subgroup of PQ. Thus we have that every228 maximal subgroup of P is a normal subgroup of PQ. Since P is an elementary abelian229 p-group and P is a non-cyclic Sylow p-subgroup, so every subgroup of order p is a normal230 subgroup in PQ, where (p, |Q|) = 1, by Lemma 7(i). On the other hand, we know that231 Ri ∩ Z(P ) ̸= 1, where (i = 1, ..., t). Let Li be subgroup of Ri ∩ Z(P ) of order p, where232 (i = 1, ..., t). Then Li is normal in P and we have Li is subnormal in G. Now, if Li = P1,233 then Li is normal in G. Also, if Li is a proper subgroup of P1, then Li is an H-subgroup234 in G. Applying Lemma 4, we get Li ⊴ G. Since Ri is a minimal normal subgroup of G,235 it follows that Li = Ri is a cyclic group of order p, for any i. Therefore, we can write236 F (H) = R1×R2×R3×· · ·×Rm, where Ri(i = 1, . . . ,m) is a normal subgroup of G of prime237 order. We have G/CG(Ri) is isomorphic to a subgroup of Aut(Ri), G/CG(Ri) is cyclic, in238 particular G/CG(Ri) is supersolvable. Hence, G/∩m i=1CG(Ri) is supersolvable. Notice that239 CG(F (H)) = ∩m i=1CG(Ri), so G/CG(F (H)) is supersolvable. The supersolvability of G/H240 and G/CG(F (H)) implies that G/(H ∩CG(F (H)) = G/CH(F (H)) is supersolvable. Since241 H is solvable, CH(F (H)) ⩽ F (H). Moreover, F (H) ⩽ CH(F (H)) as F (H) is abelian.242 Hence, F (H) = CH(F (H)), and so G/F (H) is supersolvable. Then there exists a chief243 series:244 1̄ = Gm/F (H) ⊴ Gm−1/F (H) ⊴ Gm−2/F (H) ⊴ · · · ⊴ G0/F (H) = G/F (H),245 where ((Gi−1/F (H)/(Gi/F (H))(1 ⩽ i ⩽ m) are cyclic groups of prime order. Then246 F (H) = Gm ⊴ Gm−1 ⊴ Gm−2 ⊴ · · · ⊴ G0 = G, (̇∗)247 where Gi−1/Gi ∼= ((Gi−1/F (H)/(Gi/F (H))(1 ⩽ i ⩽ m) are cyclic groups of prime order248 and Gi ⊴ G. Also, we have:249 A. S. Allehyani / Eur. J. Pure Appl. Math, 18 (2) (2025), 5993 8 of 13 1 = Gn ⊴ Gn−1 ⊴ Gn−2 ⊴ · · ·Gm+1 ⊴ Gm = F (H), (̇ ∗ ∗)250 where (Gi−1/Gi)(m+ 1 ⩽ i ⩽ n) are cyclic groups of prime order and Gi ⊴ G. Then, we251 have from (∗) and (∗∗):252 1 = Gn ⊴ · · · ⊴ Gm = F (H) ⊴ · · · ⊴ G0 = G,253 where (Gi−1/Gi)(1 ⩽ i ⩽ n) are cyclic groups of prime order and Gi ⊴ G. Hence, G is254 supersolvable, a contradiction.255 Now, we generalize Theorem 2 to the class of saturated formation as follows:256 Theorem 3. Let F be a saturated formation containing U. Suppose that G is a group257 with a solvable normal subgroup H such that G/H ∈ F. If all maximal subgroups of the258 non-cyclic Sylow subgroups of F (H) are SSH-subgroups of G, then G ∈ F.259 Proof. Assume that the claim is false and choose G to be a counterexample of minimal260 order. We aim to obtain that there is no such counterexample of G by the following steps:261 262 (1) Φ(G) ∩H = 1263 If not, Φ(G) ∩H ̸= 1 and then there exists a prime p such that p||Φ(G) ∩H|. Let P1264 be a non-cyclic Sylow p-subgroup of (Φ(G) ∩H). Clearly, P1 ⊴ G and (G/P1)/(H/P1) ∼=265 G/H ∈ F. By using similar arguments as in the second paragraph of (1) in Theorem 2,266 we can see that G/P1 ∈ F. But P1 ⩽ Φ(G), then G/Φ(G) ∈ F and, since F is saturated,267 we have G ∈ F, a contradiction. Thus Φ(G) ∩H = 1.268 269 (2) Let P be a non-cyclic Sylow p-subgroup of F (H). Then P = R1 ×R2 ×R3 × · · · ×Rt,270 where Ri(i = 1, . . . , t) are normal subgroups of G of order p.271 By (1) and Lemma 3, we have P = R1 × R2 × R3 × · · · × Rt, where Ri(i = 1, . . . , t)272 is a minimal normal subgroup of G. It is easily follows, by a similar argument to (2) in273 Theorem 2, that |Ri| = p.274 275 (3) G/F (H) ∈ F.276 From (2), Denote F (H) = R1 × R2 × R3 × · · · × Rr, where Ri(i = 1, . . . , r) are minimal277 normal subgroups of G. We have G/CG(Ri) is isomorphic to a subgroup of Aut(Ri),278 which implies that G/CG(Ri) is cyclic, and G/CG(Ri) ∈ U. So G/(∩r i=1CG(Ri)) ∈ U.279 Notice that CG(F (H)) = ∩r i=1CG(Ri), so G/CG(F (H)) ∈ U ⊆ F. Since G/H ∈ F and280 G/CG(F (H)) ∈ F, it follows that G/(H ∩ CG(F (H)) = G/CH(F (H)) ∈ F. As H is281 solvable, CH(F (H)) ⩽ F (H). Moreover, F (H) ⩽ CH(F (H)) as F (H) is abelian. Hence,282 F (H) = CH(F (H)), and so G/F (H) ∈ F.283 284 (4) If N is a minimal normal subgroup of G contained in H, then G/N ∈ F.285 Let N be an arbitrary minimal normal subgroup of G contained in H. Since H is286 solvable, we may assume that N is an elementary abelian p-group for some prime p and287 N ⩽ F (H). Now, we show G/N and F (H)/N satisfy the hypothesis of the theorem.288 Consider the solvable normal subgroup F (H)/N . Then289 A. S. Allehyani / Eur. J. Pure Appl. Math, 18 (2) (2025), 5993 9 of 13 (G/N)/(F (H)/N) ∼= G/F (H) ∈ F.290 To prove G/N ∈ F, we need only show that all maximal subgroups of the non-cyclic291 Sylow subgroups of F (H)/N = F (F (H)/N) are SSH-subgroups in G/N . Now P/N is292 the non-cyclic Sylow p-subgroup of F (H)/N , where P is the non-cyclic Sylow p-subgroup293 of F (H). Thus if P1/N is maximal in P/N , P1 is maximal in P , so P1 is an SSH-subgroup294 in G by hypothesis, and P1/N is SSH-subgroup in G/N by Lemma 2(ii). Now suppose q295 is a prime different from p, so QN/N is the Sylow q-subgroup of F (H)/N , where Q is the296 Sylow q-subgroup of F (H). Then any maximal subgroup of QN/N is of the form Q1N/N ,297 where Q1 is a maximal subgroup of Q. Thus Q1 is an SSH-subgroup in G by hypothesis,298 so Q1N/N is an SSH-subgroup in G/N by Lemma 2 (ii). So G/N and F (H)/N satisfy299 the hypotheses of the theorem. It follows that G/N ∈ F.300 301 (5) The final contradiction302 By (2) and (4), F (H) = ⟨x1⟩, is the unique minimal normal subgroup of G contained in303 H, so F (H) is cyclic of prime order. Let N = F (H), we show that N is the only minimal304 normal subgroup of G. Suppose that L ̸= N is another minimal normal subgroup of G,305 and consider NL/L normal subgroup of G/L. Since306 (G/L)/(NL/L) ∼= G/NL ∼= (G/N)/(NL/N),307 and G/N ∈ F, we have (G/L)/(NL/L) ∈ F. Notice that N ∩ L = 1, hence (NL/L) ∼= N .308 And so, the only maximal subgroup of the non-cyclic Sylow subgroup of F (NL/L) =309 NL/L is trivial subgroup, which is an SSH-subgroup in G/L. By the minimal choice of310 G, G/L ∈ F. So, G ∈ F, a contradiction. Thus, N = F (H) = ⟨x1⟩ is unique minimal311 normal in G. By (1), Φ(G) = ⟨x1⟩ ∩ Φ(G) = 1. Let M be maximal subgroup of G312 such that ⟨x1⟩ ⊈ M . Then G = ⟨x1⟩M and ⟨x1⟩ ∩ M = 1. If ⟨x1⟩ < CG(⟨x1⟩), then313 1 < CG(⟨x1⟩) ∩ M ⩽ ⟨x1⟩M = G. By the unique minimal normality of ⟨x1⟩, ⟨x1⟩ ⩽314 CG(⟨x1⟩) ∩M ⩽ M , then G = ⟨x1⟩M = M , a contradiction. Thus, ⟨x1⟩ = CG(⟨x1⟩). It315 follows that G/⟨x1⟩ = G/CG(⟨x1⟩) ⊆ Aut(⟨x1⟩) is cyclic of order dividing p − 1 and so316 G/⟨x1⟩) ∈ U. Hence, G ∈ U ⊆ F, the final contradiction.317 In the following remark, we will mention some cases.318 Remark 1. (i) Theorem 3 is not true if we omit the solvability of H. Set G = N ×M ,319 where N = SL(2, 5), the special linear group of degree 2 and M ∈ U. Then F (N) =320 Z(N) ∼= Z2 and G/N ∼= M ∈ U, but G does not belong to U.321 (ii) Theorem 3 is not true for saturated formations F which do not contain U. For322 example, if F is the saturated formation of all nilpotent groups, then the symmetric323 group S3 of degree three is a counterexample.324 Theorem 4. Let G be a group with a normal subgroup H such that G/H is supersolvable.325 If all maximal subgroups of the non-cyclic Sylow subgroups of F ∗(H) are SSH-subgroups326 of G, then G is supersolvable.327 A. S. Allehyani / Eur. J. Pure Appl. Math, 18 (2) (2025), 5993 10 of 13 Proof. Suppose that the theorem is false and assume that G is a counterexample of328 minimal order. Then we have:329 330 (1) Every proper normal subgroup of G containing F ∗(H) is supersolvable.331 If N is a proper normal subgroup of G containing F ∗(H), we have N/N ∩H ∼= NH/H is332 supersolvable as NH/H ⩽ G/H which is supersolvable. By Lemma 8((i) and (ii)),333 F ∗(H) = F ∗(F ∗(H)) ⩽ F ∗(H ∩N) ⩽ F ∗(H),334 so, F ∗(H) = F ∗(H ∩N). Then all maximal subgroups of the non-cyclic Sylow subgroups335 of F ∗(H ∩ N) (i. e. of F ∗(H)) are SSH-subgroups in G. Thus, all maximal subgroups336 of the non-cyclic Sylow subgroups of F ∗(H ∩N) (i. e. of F ∗(H)) are SSH-subgroups in337 N by Lemma 2(ii). So, we have N , H ∩ N satisfy the hypothesis of the theorem. The338 minimality of G implies that N is supersolvable.339 340 (2) H = G, and F ∗(H) = F (H) < G341 If H < G, then H is supersolvable by (1). In particular, H is solvable, so by Lemma 9, G342 is solvable and F ∗(H) = F (H) by Lemma 8(ii), then G is supersolvable by Theorem 2, a343 contradiction.344 If F ∗(H) = G, then G is supersolvable by Theorem 3, a contradiction. Then F ∗(H) < G345 and it is supersolvable by (1). So, F ∗(H) = F (H).346 347 (3) For any Sylow p-subgroup P of F (G), Φ(P ) = 1, i.e. P is elementary abelian.348 If there exists a Sylow p-subgroup P of F (G) with Φ(P ) ̸= 1, then consider the factor349 group G/Φ(P ). By Lemma 8(iii), F ∗(G/Φ(P )) = F ∗(G)/Φ(P ) = F (G)/Φ(P ). If P1/Φ(P )350 is a maximal subgroup of the non-cyclic Sylow p-subgroup P/Φ(P ) of F ∗(G)/Φ(P ), then351 P1 is a maximal subgroup of the non-cyclic Sylow p-subgroup P of F ∗(G). So, P1 is an352 SSH-subgroup, by hypothesis. Then P1/Φ(P ) is an SSH-subgroup by Lemma 2(iii). If353 Q∗/Φ(P ) is a maximal subgroup of the non-cyclic Sylow q-subgroup of QΦ(P )/Φ(P ) of354 F ∗(G)/Φ(P ), where Q is the non-cyclic Sylow q-subgroup of F ∗(G) and q ̸= p, then we355 can denoted Q∗ = Q1Φ(P ), where Q1 is a maximal subgroup of the non-cyclic Sylow356 q-subgroup of Q of F ∗(G). Now, Q1 is an SSH-subgroup (by hypothesis and by Lemma357 2 (iii)), implies that Q∗/Φ(P ) is an SSH-subgroup in G/Φ(P ). By minimality of G,358 G/Φ(P ) is supersolvable. But P ⊴ G, then Φ(P ) ⩽ Φ(G), and we get G/Φ(G) is super-359 solvable. By Huppert’s Theorem [13, p. 713, Satz 8.6], G is supersolvable, a contradiction.360 361 (4) There is no subgroup of prime order normal in G.362 If not, let P0 be a normal subgroup of G of prime order p. Then P0 ⩽ P as P ⊴ G. Since363 P0 ⩽ Z(P ) ⩽ Z(F (G)), it follows that F (G) ⩽ CG(P0) ⩽ G. By (2) and Lemma 8((i)364 and (ii)), F ∗(G) ⩽ F ∗(CG(P0)). But F ∗(CG(P0)) ⩽ F ∗(G). Therefore, by the fact that365 CG(P0) ⊴ G and Lemma 8(i), F ∗(CG(P0)) = F ∗(G) = F (G). If further CG(P0) < G, then366 CG(P0) is supersolvable by (1). Since G/CG(P0) is isomorphic to a subgroup of Aut(P0),367 which is cyclic, we get that G/CG(P0) is cyclic and hence solvable. But G/CG(P0) is368 solvable, then G is solvable. Applying Theorem 1 implies that G is supersolvable, a con-369 A. S. Allehyani / Eur. J. Pure Appl. Math, 18 (2) (2025), 5993 11 of 13 tradiction. If CG(P0) = G, then P0 ⩽ Z(G). By Lemma 8(iii), F ∗(G/P0) = F ∗(G)/P0. By370 using similar argument in (3), we get that all maximal subgroups of the non-cyclic Sylow371 subgroups of F ∗(G/P0) are SSH-subgroups in G/P0. The minimal choice of G implies372 that G/P0 is supersolvable. Therefore, by Lemma 9, G is supersolvable, a contradiction.373 374 (5) The Final contradiction.375 From (3), Φ(P ) = 1. Then by Lemma 3, F (G) is a direct product of minimal normal376 subgroups of G which are contained in F (G). Let P be a non-cyclic Sylow p-subgroup of377 F (G). Then P = R1 × R2 × R3 × · · · × Rt, where Ri(i = 1, . . . , t) is a minimal normal378 subgroup of G. Let P be a non-cyclic Sylow p-subgroup of F (G) and P is characteristic379 in F (G) ⊴ G, then P ⊴ G. Then there exists a maximal subgroup P1 of P , and by380 hypothesis, P1 is an SSH-subgroup in G. Then there exists an S-permutable subgroup381 K of G such that (P1) SG = P1K and (P1) g ∩NK(P1) ⩽ P1, for all g ∈ G. Assuming that382 K = P , then we have (P1) g∩NG(P1) = (P1) g∩K∩NG(P1) = (P1) g∩NK(P1) ⩽ P1. Then383 we get P1 is an H-subgroup in G and P1 ⊴ P . Applying Lemma 4, we get P1 ⊴ G. Let384 Q be a non-cyclic Sylow q-subgroup of F (G) such that (p, |Q|) = 1. Now, P1Q ⩽ G and385 since P1 is a normal Hall subgroup of P1Q, it follows that P1 is a characteristic subgroup386 of P1Q. In particular P1 is a normal subgroup of P1Q. Hence Q ⩽ NG(P1) for all Sylow387 q-subgroup Q of F (G), where (p, |Q|) = 1. Since P1 is a normal subgroup of P and P1 is388 a normal subgroup of P1Q, we get P1 is a normal subgroup of PQ. Thus we have that389 every maximal subgroup of P is a normal subgroup of PQ by Lemma 7(ii). Since P is an390 elementary abelian p-group and P is a non-cyclic Sylow p-subgroup, so every subgroup of391 order p is a normal subgroup in PQ, where (p, |Q|) = 1. On the other hand, we know that392 Ri ∩ Z(P ) ̸= 1, where (i = 1, ..., t). Let Li be subgroup of Ri ∩ Z(P ) of order p, , where393 (i = 1, ..., t). Then Li is normal in P and we have Li is subnormal in G. Now, if Li = P1,394 then Li is normal in G. Also, if Li is a proper subgroup of P1, then Li is an H-subgroup395 in G. Applying Lemma 4, we get Li ⊴ G. Since Ri is a minimal normal subgroup of G,396 it follows that |Li| = |Ri| = p is a cyclic group of order p, for any i, which contradict (4)397 completing the proof of the theorem.398 As an application of Theorem 4, we have:399 Theorem 5. Let F be a saturated formation containing U. A group G lies in F if and400 only if it has a solvable normal subgroup H such that G/H ∈ F and all maximal subgroups401 of the non-cyclic Sylow subgroups of F ⋆(H) are SSH-subgroups in G.402 Proof. We need only to prove the part ”if”. We use induction on |G|. By hypothesis403 and Lemma 2(i), we have that all maximal subgroups of the non-cyclic Sylow subgroups404 of F ⋆(H) are SSH-subgroups of H. Then F ⋆(H) = F (H) as H is supersolvable by405 Theorem 4. Therefore, H is solvable normal subgroup of G with G/H ∈ F and all406 maximal subgroups of the non-cyclic Sylow subgroups of F (H) are SSH-subgroups in G.407 Applying Theorem 3 yield G ∈ F. This completes the proof of the theorem.408 Remark 2. (i) Theorem 5 is not true if we omit the solvability of H. Set G = N ×M ,409 A. S. Allehyani / Eur. J. Pure Appl. Math, 18 (2) (2025), 5993 12 of 13 where N = SL(2, 5), the special linear group of degree 2 and M ∈ U. Then F ⋆(N) =410 N and G/N ∼= M ∈ U, but G does not belong to U.411 (ii) Theorem 5 is not true for non-saturated formation. For example, let F be the forma-412 tion composed of all groups G such that GU, the supersolvable residual, is elementary413 Abelian. It is clear that U ⊆ F and F is not a saturated formation. Let G = SL(2, 3)414 and N = Z(G). Then G/N ∼= A4, so G/N ∈ F. But G does not belong to U.415 4. Conclusion416 Due to the importance of finite groups theory and its application in abstract algebra,417 our study in this article focused on the structure of a finite group G assuming that some418 subgroups of prime power order are SSH-subgroups. In the current article, we have419 reached the following results: If G is solvable and the maximal subgroups of the non-420 cyclic Sylow subgroups of F (G) are SSH-subgroups, then G is supersolvable. Also, let421 G be a group with a normal subgroup H such that G/H is supersolvable. If all maximal422 subgroups of the non-cyclic Sylow subgroups of F ⋆(H) are SSH-subgroups of G, then G423 is supersolvable. Finally, several recent and classical results were generalized through the424 theory of formations.425 Acknowledgements426 The author thank the reviewers for their valuable and helpful suggestions and com-427 ments.428 References429 [1] O. H. Kegel. Sylow-Gruppen und Subnormalteiler endlicher Gruppen. Mathematische430 Zeitschrift, 78(1):205–221, 1962.431 [2] Y. Wang. c-normality of groups and its properties. Journal of Algebra, 180(3):954–432 965, 1996.433 [3] M. Bianchi, A. G. B. Mauri, M. Herzog, and L. Verardi. On finite solvable groups434 in which normality is a transitive relation. Journal of Group Theory, 3(2):147–156,435 2000.436 [4] X. Wei and X. Guo. On HC-subgroups and the structure of finite groups. Communi-437 cations in Algebra, 40(9):3245–3256, 2012.438 [5] M. Asaad and M. Ramadan. On weakly HC-embedded subgroups of finite groups.439 Journal of Algebra and its Applications, 15(5):1650091, 2016.440 [6] T. M. Al-Gafri and S. K. Nauman. On SSH-subgroups of finite groups. Annali441 dell’Università di Ferrara, 64(2):209–225, 2018.442 [7] S. Srinivasan. Two sufficient conditions for supersolvability of finite groups. Israel443 Journal of Mathematics, 35(3):210–214, 1980.444 A. S. Allehyani / Eur. J. Pure Appl. Math, 18 (2) (2025), 5993 13 of 13 [8] M. Asaad. On p-nilpotence and supersolvability of finite groups. Communications in445 Algebra, 34(1):189–195, 2006.446 [9] M. M. Al-Shomrani, M. Ramadan, and A. A. Heliel. Finite groups whose minimal447 subgroups are weakly H-subgroups. Acta Mathematica Scientia, 32(6):2295–2301,448 2012.449 [10] M. Asaad, M. M. Al-Shomrani, and A. A. Heliel. Influence of weakly H-subgroups450 on the structure of finite groups. Studia Scientiarum Mathematicarum Hungarica,451 51(1):27–40, 2014.452 [11] M. Asaad, M. Ramadan, and A. A. Heliel. Influence of weaklyH-embedded subgroups453 on the structure of finite groups. Publicationes Mathematicae Debrecen, 91(3-4):503–454 513, 2017.455 [12] M. Asaad, A. A. Heliel, and M. M. Al-Shomrani. On weakly H-subgroups of finite456 groups. Communications in Algebra, 40(9):3540–3550, 2012.457 [13] B. Huppert. Endliche Gruppen I. Springer, Berlin, 1967.458 [14] W. Guo. The theory of classes of groups. Kluwer Academic Publishers, Dordrecht,459 2000.460 [15] D. J. S. Robinson. A course in the theory of groups. Springer, New York, 1993.461 [16] L. Miao and W. Lempken. On M-supplemented subgroups of finite groups. Journal462 of Group Theory, 12(2):271–287, 2009.463 [17] Y. Wang, Y. Li, and H. Wei. The influence of π-quasinormality of some subgroups of464 a finite group. Archiv der Mathematik, 81(3):245–252, 2003.465 [18] W. Guo and A. N. Skiba. Finite groups with given S-embedded and n-embedded466 subgroups. Journal of Algebra, 321(10):2843–2860, 2009.467 [19] M. Asaad and A. A. Heliel. On permutable subgroups of finite groups. Archiv der468 Mathematik, 80(2):113–118, 2003.469 [20] B. Huppert and N. Blackburn. Finite groups III. Springer, Berlin, 1982.470 [21] M. Weinstein. Between nilpotent and solvable. Polygonal Publishing House, Passaic,471 NJ, 1982.472