ISSN 0001-4346, Mathematical Notes, 2014, Vol. 95, No. 2, pp. 267–276. © Pleiades Publishing, Ltd., 2014. Published in Russian in Matematicheskie Zametki, 2014, Vol. 95, No. 2, pp. 300–311.

On SS-Quasinormal and S-Quasinormally Embedded Subgroups of Finite Groups* Zhencai Shen1** , Shirong Li2 , and Jinshan Zhang3 1

China Agricultaral University, Peking, China 2 Guangxi University, China 3 Sichuan University of Science and Engineering, China Received April 8, 2010

Abstract—A subgroup H of a group G is said to be an SS-quasinormal (SupplementSylow-quasinormal) subgroup if there is a subgroup B of G such that HB = G and H permutes with every Sylow subgroup of B. A subgroup H of a group G is said to be S-quasinormally embedded in G if for every Sylow subgroup P of H, there is an S-quasinormal subgroup K in G such that P is also a Sylow subgroup of K. Groups with certain SS-quasinormal or S-quasinormally embedded subgroups of prime power order are studied. DOI: 10.1134/S0001434614010283 Keywords: SS-quasinormal subgroup, p-nilpotent group, supersolvable group, formation.

1. INTRODUCTION All groups considered in this paper will be finite; the notation and terminology used in this paper are standard, as in [1]–[4]. Given a group G, two subgroups H and K of G are said to permute if HK = KH, that is, HK is a subgroup of G. A subgroup H of a group G is said to be S-permutable in G (or S-quasinormal in G) if H permutes with every Sylow subgroup of G. This concept was introduced by Kegel and Deskins in 1962 and has been investigated by many authors, for example, see [5]–[23]. In 1998, the authors of [6] extended this concept to S-quasinormally embedded subgroups. Definition 1.1. A subgroup H of a group G is said to be S-quasinormally embedded in G if for every Sylow subgroup P of H, there is an S-quasinormal subgroup K in G such that P is also a Sylow subgroup of K. Recall that a formation is a class F of groups satisfying the following conditions: (i) if G ∈ F and N  G, then G/N ∈ F; (ii) if N1 , N2  G and G/N1 , G/N2 ∈ F, then G/(N1 ∩ N2 ) ∈ F. A formation F is said to be saturated if G/Φ(G) ∈ F implies that G ∈ F, see [4]. Recently, in 2007, Skiba [20] introduced the concept of weakly s-permutable subgroup and established an interesting theorem on supersolvable groups. He proved the following result. Let F be a saturated formation containing all supersolvable groups, and let G be a group with normal subgroup E such that G/E ∈ F. Suppose that every noncyclic Sylow subgroup P of F ∗ (E) has a subgroup D such that 1 < |D| < |P | and all subgroups H of P of order |H| = |D| and of order 2|D| (if P is a non-Abelian 2-group and |P : D| > 2) are weakly s-permutable in G. Then G ∈ F. In 2008, Li, Shen and other authors gave the following definition. ∗ **

The text was submitted by the authors in English. E-mail: [email protected], [email protected]

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Definition 1.2. Let G be a group. A subgroup H of G is said to be an SS-quasinormal subgroup (supplement-Sylow-quasinormal subgroup) of G if there is a supplement B of H in G such that H permutes with every Sylow subgroup of B. Obviously, every S-quasinormal subgroup of a group G is SS-quasinormal and S-quasinormally embedded in G. In general, an SS-quasinormal subgroup need not be S-quasinormally embedded. For instance, S3 is an SS-quasinormal subgroup of the symmetric group S4 , but S3 is not S-quasinormally embedded and so is not S-quasinormal. The converse is also true, for example, a Sylow 3-subgroup of A5 is S-quasinormally embedded but not SS-quasinormal. In fact, there is no inclusion relationship between the two concepts. Definition 1.3. Let d be the smallest generator number of a p-group P and Md (P ) = {P1 , . . . , Pd } be  a set of maximal subgroups of P such that di=1 Pi = Φ(P ). Such a subset Md (P ) is not unique for a fixed P in general. We know that |M(P )| =

pd − 1 , p−1

|Md (P )| = d,

(pd − 1)/(p − 1) = ∞, d→∞ d lim

so |M(P )|  |Md (P )|. In this paper, we study the influence of SS-quasinormal or S-quasinormally embedded subgroups on the structure of the group G. The main results are as follows. Theorem 1.1. Let p be the smallest prime dividing the order of a group G, and let P be a Sylow p-subgroup of G. Then G is p-nilpotent if and only if every maximal subgroup of P not having a p-nilpotent supplement is SS-quasinormal in G. Theorem 1.2. Let p be the smallest prime dividing the order of a group G, and let P be a Sylow p-subgroup of G. Then the following statements are equivalent: (i) G is p-nilpotent; (ii) if P has a subgroup D such that 1 < |D| < |P | and all subgroups H of P of order |H| = |D| and of order 2|D| (if P is a non-Abelian 2-group and |P : D| > 2) not having a p-nilpotent supplement in G are SS-quasinormal in G. Theorem 1.3. Let F be a saturated formation containing all supersolvable groups, and let G be a group with normal subgroup E such that G/E ∈ F. Suppose that every noncyclic Sylow subgroup P of F ∗ (E) has a subgroup D such that 1 < |D| < |P | and all subgroups H of P of order |H = |D| and of order 2|D| (if P is a non-Abelian 2-group and |P : D| > 2) not having a supersolvable supplement in G are SS-quasinormal in G. Then G ∈ F. Theorem 1.4. Let p be the smallest prime dividing the order of a group G of odd order, and let P be a Sylow p-subgroup of G. Then G is p-nilpotent if and only if every maximal subgroup of P not having p-nilpotent supplement is S-quasinormally embedded in G. Theorem 1.5. Let p be the smallest prime dividing the order of a group G of odd order, and let P be a Sylow p-subgroup of G. Then the following statements are equivalent: (i) G is p-nilpotent; (ii) if P has a subgroup D such that 1 < |D| < |P | and all subgroups H of P of order |H| = |D| not having a p-nilpotent supplement in G are S-quasinormally embedded in G. Theorem 1.6. Let F be a saturated formation containing all supersolvable groups, and let G be a group of odd order with a normal subgroup E such that G/E ∈ F. Suppose that every noncyclic Sylow subgroup P of F ∗ (E) has a subgroup D such that 1 < |D| < |P | and all subgroups H of P of order |H| = |D| not having a supersolvable supplement in G are S-quasinormally embedded in G. Then G ∈ F. MATHEMATICAL NOTES

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Remark. We do not know whether Theorems 1.4, 1.5, and 1.6 are true or not if we omit the condition that G is a group of odd order. In [11], [16], [17], we obtained the following theorems. Theorem 1.7. Let p be the smallest prime dividing the order of a group G, and let P be a Sylow p-subgroup of G. Suppose that every member of some fixed Md (P ) is SS-quasinormal in G, then G is p-nilpotent. Theorem 1.8. Let p be the smallest prime dividing the order of a group G, and let P be a Sylow p-subgroup of G. Then the following two statements are equivalent: (i) G is p-nilpotent; (ii) every member of some fixed Md (P ) is S-quasinormally embedded in G. Theorem 1.9. Let p be the smallest prime dividing the order of a group G, and let P a Sylow p-subgroup of G. Then the following two statements are equivalent: (i) G is p-nilpotent; (ii) every member of some fixed Md (P ) is S-quasinormally embedded or SS-quasinormal in G. However, the following statements are not true. Conjecture 1.10. Let p be the smallest prime dividing the order of a group G, and let P be a Sylow p-subgroup of G. Suppose that every member of some fixed Md (P ) not having a p-nilpotent supplement is SS-quasinormal in G, then G is p-nilpotent. Conjecture 1.11. Let p be the smallest prime dividing the order of a group G, and let P be a Sylow p-subgroup of G. Suppose that every member of some fixed Md (P ) not having a p-nilpotent supplement is S-quasinormally embedded in G, then G is p-nilpotent. For example, the symmetric group S4 is not 2-nilpotent and is such that P = D8 is a Sylow 2-subgroup of G and every member of some fixed Md (P ) not have 2-nilpotent supplement is S-quasinormally embedded (SS-quasinormal) in G. 2. PRELIMINARIES Lemma 2.1 ([11]). Suppose that H is SS-quasinormal in a group G, K ≤ G and N is a normal subgroup of G. Then (i) if H ≤ K, then H is SS-quasinormal in K; (ii) HN/N is SS-quasinormal in G/N ; (iii) if N ≤ K and K/N is SS-quasinormal in G/N , then K SS-quasinormal in G. Lemma 2.2 ([11]). Let H be a p-subgroup of G. Then the following statements are equivalent: i) H is S-permutable in G; ii) H ≤ Op (G) and H is S-quasinormally embedded in G; iii) H ≤ Op (G) and H is SS-quasinormal in G. Lemma 2.3 ([6]). Suppose that H is S-quasinormally embedded in a group G, K ≤ G, and N is a normal subgroup of G. Then: MATHEMATICAL NOTES

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(i) if H ≤ K, then H is S-quasinormally embedded in K; (ii) HN/N is S-quasinormally embedded in G/N . By Lemma 2.11 [20] and Lemma 2.2, we have the following. Lemma 2.4. 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 SS-quasinormal or S-quasinormally embedded in G. Then some maximal subgroup of N is normal in G. Lemma 2.5. Let F be a saturated formation containing all nilpotent groups, and let G be a group with solvable F-residual P = GF . Suppose that every maximal subgroup of G not containing P belongs to F. Then P is a p-group for some prime p. In addition, if every cyclic subgroup of P with prime order or order 4 (if p = 2 and P is non-Abelian) not having a supersolvable supplement in G is SS-quasinormal or S-quasinormally embedded in G, then |P/Φ(P )| = p. Lemma 2.6 ([4]). Let G be a group, and let M be a subgroup of G. Then the following statements are equivalent: (i) if M is normal in G, then F ∗ (M ) ≤ F ∗ (G); (ii) F ∗ (G) = 1 if G = 1; in fact, F ∗ (G)/F (G) = soc(F (G)CG (F (G))/F (G)); (iii) F ∗ (F ∗ (G)) = F ∗ (G) ≥ F (G); if F ∗ (G) is solvable, then F ∗ (G) = F (G); iv) if K is a subgroup of G contained in Z(G), then F ∗ (G/K) = F ∗ (G)/K. Lemma 2.7 ([3]). Let A and B be subgroups of a group G satisfying G = AB. If AB g = B g A holds for all g ∈ G, then A or B is contained in a proper normal subgroups of G. By [20, Lemma 2.10], we have the following lemma. Lemma 2.8. Let p be the smallest prime dividing the order of a group G, and let P be a Sylow p-subgroup of G. Then G is p-nilpotent if and only if P has a subgroup D such that 1 < |D| < |P | and every subgroup H of P of order |H| = |D| and of order 2|D| (if P is a non-Abelian 2-group and |P : D| > 2) has p-nilpotent supplement in G. Lemma 2.9 ([5]). Let G be a group and K a p-subgroup, where p is a prime. If K is an S-quasinormally embedded subgroup of G and KG = 1, then K is S-quasinormal in G. Lemma 2.10. Let N  G and P be a p-subgroup of a group G, where p is a prime dividing the order of group G. If P is SS-quasinormal in G, then P ∩ N is permutable with all Sylow q-subgroup Q of N , for all q = p. Proof. Let Q be a Sylow q-subgroup of N with q = p. Then there exists a Sylow q-subgroup Gq of G such that Q ≤ Gq . Since P is SS-quasinormal in G, P Gq is a subgroup and P Gq ∩ N Gq = (P ∩ N Gq )Gq = (P ∩ N )Gq is also a subgroup. Moreover, (P ∩ N )Gq ∩ N = (P ∩ N )(Gq ∩ N ) = (P ∩ N )Q. Thus, (P ∩ N )Q is a subgroup and so (P ∩ N )Q = Q(P ∩ N ), as desired. MATHEMATICAL NOTES

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3. PROOFS OF THE MAIN THEOREMS ON SS-QUASINORMALITY Proof of Theorem 1.1. The sufficiency is obvious. We prove the necessity. Suppose the theorem is false and G is a counter-example of minimal order. We will derive a contradiction in several steps. (1) G has a unique minimal normal subgroup N such that G/N is p-nilpotent and Φ(G)=1. Let N be a minimal normal subgroup of G. By Lemma 2.1, G/N satisfies the hypothesis of the theorem. The choice of G implies that G/N i s p-nilpotent. Moreover, N is the unique minimal normal subgroup and Φ(G) = 1. (2) Op (G)=1. If Op (G) = 1, Then N ≤ Op (G) by (1). By Lemma 2.1, G/N satisfies the hypothesis. Thus, G/N is p-nilpotent. Now the p-nilpotency of G/N implies that G is p-nilpotent; a contradiction. (3) Op (G)=1. If Op (G) = 1, then N ≤ Op (G) by (1). Therefore, G has a maximal subgroup M such that G = M N and M ∩ N = 1. By (1) we have Φ(Op (G)) = 1. Thus, Op (G) ∩ M is normalized by N and M . Hence, by the uniqueness of N , we have N = Op (G). By Lemma 2.8, there is a maximal subgroup P1 of P such that P1 does not have p-nilpotent supplement. Since N has a p-nilpotent supplement M , we have N  P1 . Moreover, by the hypotheses, P1 is SS-quasinormal in G. Therefore, P1 Gq is a subgroup, where q = p and Gq ∈ Sylq (G). Since P1 ∩ N = P1 Gq ∩ N  P1 Gq , we have Gq ≤ NG (P1 ∩ N ). Moreover, P1 ∩ N  P . Therefore, P1 ∩ N  G. By the uniqueness of N , we have P1 ∩ N = 1, and so |N | = p. The p-nilpotenty of M implies that G is p-nilpotent; a contradiction. Thus, (3) holds. (4) G is nonsolvable and hence N is a direct product of some non-Abelian simple groups. This follows from (2) and (3). (5) The final contradiction Let H be a maximal subgroup of P . By the hypothesis, H is SS-quasinormal or has p-nilpotent supplement in G. If H is SS-quasinormal, by Lemma 2.10, then H ∩ N permutes with all Sylow q-subgroups Q of N , for all q = p. Since N is a direct product of some non-Abelian simple groups, by Lemma 2.7 and [3, I, Satz 9.12], there is a proper normal subgroup M of N such that H ∩ N ≤ M . Thus, |N/M |p = p, and so N/M is p-nilpotent, which contradicts (4). So every maximal subgroup of P has a p-nilpotent supplement and it follows by Lemma 2.8 that G is p-nilpotent; a contradiction. The proof of the theorem is complete. Proof of Theorem 1.2. It is clear that (i) implies (ii). Next, we prove that (ii) implies (i). Assume that the theorem is not true and let G be a counterexample of minimal order. We prove the theorem in the following several steps. (1) Op (G) = 1. In fact, if Op (G) = 1, then we consider the quotient group G/Op (G). By Lemma 2.1, G/Op (G) satisfies the hypotheses of the theorem, and it follows that G/Op (G) is p-nilpotent by the choice of G. Hence G is p-nilpotent; a contradiction. (2) |D| > p. If |D| = p, then by Lemma 2.1, G is a minimal non-p-nilpotent group, so G = [P ]Q, where P and Q are Sylow p-subgroup and Sylow q-subgroup of G, respectively. Set Φ = Φ(P ), and let X/Φ be a subgroup of P/Φ of order p, x ∈ X \ Φ and L = x . Then L is of order p or 4. By the hypotheses, L has a p-nilpotent supplement in G or is SS-quasinormal in G. If L has a p-nilpotent supplement T in G, then T = G. So |G/Φ : T Φ/Φ| = p. Hence T Φ/Φ  G/Φ

and

P/Φ ∩ T Φ/Φ = 1,

and it follows that |P/Φ| = p. Therefore, P is cyclic and G is p-nilpotent; a contradiction. So L is SS-quasinormal in G. By Lemma 2.2, L is S-permutable in G. Lemma 2.5 implies that |P/Φ| = p, and it follows that G is p-nilpotent. (3) |P : D| > p. If |P : D| = p, then we see that G is p-nilpotent by Theorem 1.1; a contradiction. (4) If N ≤ P and N is a minimal normal subgroup of G, then |N | ≤ |D|. Let |N | > |D|. Since N ≤ Op (G), N is an elementary Abelian group. If a subgroup H of N of order |D| has a p-nilpotent supplement T in G, then G = HT = N T . Thus, N ∩ T  G. By the minimality of N , we have N ∩T = 1 MATHEMATICAL NOTES

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N ∩ T = N.

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If N ∩ T = 1, then N = N ∩ HT = H(N ∩ T ) = H; a contradiction. Therefore, N ∩T =N

and

G = NT = T;

this is also a contradiction. Hence all subgroups of N of order |D| are SS-quasinormal in G. By Lemma 2.4, some maximal subgroup N1 of N is normal in G. It follows from the minimality of N that N1 = 1, thus |N | = |D| = p; a contradiction. (5) If N ≤ P and N is minimal normal subgroup of G, then G/N is p-nilpotent. If |N | < |D|, then G/N is p-nilpotent by Lemma 2.1. By (4), we have |N | = |D|. Let N ≤ K ≤ P and |K/N | = p. By (2) N is noncyclic, so K is also noncyclic, it follows that K has a maximal subgroup L = N and K = LN . If L has a p-nilpotent supplement in G, then K has a p-nilpotent supplement in G. If L is SS-quasinormal in G, it follows that K/N = LN/N is SS-quasinormal in G/N by lemma 2.1. If P/N is Abelian, then G/N satisfies the hypothesis. Next, suppose that that P/N is a non-Abelian 2-group. So every subgroup of P of order 2|D| not having a p-nilpotent supplement in G is SS-quasinormal in G. In this case one can show as above that every subgroup X of P containing N and such that |X : N | = 4 either has a p-nilpotent supplement in G or is SS-quasinormal in G. Therefore, G/N also satisfies the hypothesis. (6) Op (G) = 1. If Op (G) = 1, then we can find a minimal normal subgroup N of G contained in Op (G). By the hypothesis, we have N = Op (G). So there is a maximal subgroup M of G such that G = N M , M ∩ N = 1. By (5), M is p-nilpotent. So M = Mp Op (M )

and

G = N Mp Op (M ).

Let M0 be a maximal subgroup of Mp . Then |G : (N M0 Op (M ))| = p. Since p is the smallest prime, we have N M0 Op (M )  G. Moreover, by (2) and the condition, N M0 Op (M ) is p-nilpotent. So Op (M )  G

and

Op (M ) = Op (G).

Hence G is p-nilpotent and we have (6). (7) G is non-Abelian and simple. If not, then there exists a minimal normal subgroup L. If |Lp | > |D|, then by the hypothesis, L is p-nilpotent. By (1), Op (G) = 1; so L is a p-group and (6) implies L = 1; this is a contradiction. Therefore, |Lp | ≤ |D|. So there exists a subgroup P0 such that P0 ≥ L ∩ P and |P0 | = p|D|. Moreover, we see that P0 is a Sylow p-subgroup of P0 L. By Theorem 1.1, P0 L is p-nilpotent. Hence L is p-nilpotent; a contradiction. (8) The final contradiction. Let H be a subgroup of P of order |D|. If H is SS-quasinormal in G, then there exists a Sylow q-subgroup Q of G such that HQg = Qg H, where q = p and g ∈ G. By Lemma 2.7, we have G = HQ, which contradicts (7). So every subgroup H of P of order |H| = |D| and of order 2|D| (if P is a nonAbelian 2-group and |P : D| > 2) has p-nilpotent supplement in G, it follows by Lemma 2.8 that G is p-nilpotent; a contradiction. The proof of the theorem is complete. Theorem 3.1. Let F be a saturated formation containing all supersolvable groups and G a group with a normal subgroup E such that G/E ∈ F. Suppose that every noncyclic Sylow subgroup P of E has a subgroup D such that 1 < |D| < |P | and all subgroups H of P of order |H| = |D| and of order 2|D| (if P is a non-Abelian 2-group and |P : D| > 2) not having a supersolvable supplement in G are SS-quasinormal in G. Then G ∈ F. Proof. Suppose that the theorem is not true and let G be a counter-example of the smallest order. We have the following claims. (1) G/Q ∈ F, where Q is a Sylow q-subgroup of E and q is the largest prime dividing |E|. By Lemma 2.1 and Theorem 1.2, E has the Sylow Tower property. Let q be the largest prime dividing |E|, and let Q be a Sylow q-subgroup of E. The fact that E possesses an order Sylow Tower property implies that Q is normal in E. Now Q is characteristic in E and E  G, so Q  G. MATHEMATICAL NOTES

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Furthermore, (G/Q)/(E/Q) ∼ = G/E ∈ F and Lemma 2.1 shows that G/Q satisfies the conditions of the theorem, thus by the choice of G, we have G/Q ∈ F. (2) Every subgroup H of Q of order |H| = |D| not having a supersolvable supplement in G is Spermutable in G. By Lemma 2.2, we have (2). (3) If N ≤ Q and N is a minimal normal subgroup of G, then G/N ∈ F. If either |N | < |D| or |Q : D| = q, then it is clear. So let |N | = |D| and |Q : D| > q. Let N ≤ K ≤ P , where |K/N | = p. By Lemma 2.5, |D| > q, it follows that N is noncyclic, so K is also noncyclic. Hence K has a maximal subgroup L = N and K = LN . If L has a supersolvable supplement in G, then K has a supersolvable supplement in G. If L is S-permutable in G, it follows that K/N = LN/N is S-permutable in G/N . Therefore, G/N satisfies the hypothesis, as desired. (4) Final contradiction. Let N be a minimal normal subgroup of G contained in Q. Then by (3), N is the only minimal normal subgroup of G contained in Q and so N = Q. But by Lemma 2.4, this is impossible, because Q is a minimal normal subgroup of G. This contradiction completes the proof of this theorem. By Theorem 1.3 in [20] and Lemma 2.2, we have the following result. Theorem 3.2. Let F be a saturated formation containing all supersolvable groups, and let G be a group with a solvable normal subgroup E such that G/E ∈ F. Suppose that every noncyclic Sylow subgroup P of F (E) has a subgroup D such that 1 < |D| < |P | and all subgroups H of P of order |H| = |D| and of order 2|D| (if P is a non-Abelian 2-group and |P : D| > 2) not having a supersolvable supplement in G are SS-quasinormal in G. Then G ∈ F. Theorem 3.3. Let G be a group with a normal subgroup E such that G/E is supersolvable. Suppose that every noncyclic Sylow subgroup P of F ∗ (E) has a subgroup D satisfying the inequalities 1 < |D| < |P | and all subgroups H of P of order |H| = |D| and of order 2|D| (if P is a non-Abelian 2-group and |P : D| > 2) not having a supersolvable supplement in G are SS-quasinormal in G. Then G is supersolvable. Proof. Suppose that the theorem is false and let G be a counter-example of the smallest order. Then we shall divide the proof into several steps and arrive at a contradiction. (1) Every proper normal subgroup of G containing F ∗ (E) is supersolvable. If N is a proper normal subgroup of G containing F ∗ (E), then we see that N/N ∩ E = N E/E is supersolvable. By Lemma 2.6, F ∗ (E) = F ∗ (F ∗ (E)) ≤ F ∗ (E ∩ N ) ≤ F ∗ (E), so F ∗ (E ∩ N ) = F ∗ (E). By Lemma 2.1, (N, N ∩ E) satisfies the hypotheses of the theorem, thus the minimality of the choice of G implies that N is supersolvable. (2) E = G and F ∗ (E) = F (G) < G. If E < G, then E is supersolvable by (1). In particular, E is solvable, so G is solvable and we have F ∗ (E) = F (E) by Lemma 2.6, It then follows that G is supersolvable by applying Theorem 3.1; a contradiction. If F ∗ (G) = G, then G is supersolvable by Theorem 3.1; a contradiction. Thus, F ∗ (E) < G and F ∗ (E) is supersolvable by (1), and so it follows that F ∗ (E) = F ∗ (G) = F (G) by Lemma 2.6. (3) The final contradiction. Let P be a Sylow p-subgroup of F (G) for some prime p, and let P1 be an arbitrary subgroup of P of order |D| and P1  P  F (G)  G. By the hypotheses, P1 not having a supersolvable supplement in G is SS-quasinormal in G. So P1 not having a supersolvable supplement in G is S-permutable in G by Lemma 2.2. Thus, all subgroups of P of order |D| not having a supersolvable supplement in G are S-permutable in G. Applying Theorem 3.1, we conclude that G is supersolvable; the final contradiction.

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Proof of Theorem 1.3. By Lemma 2.1, we see that all subgroups of any Sylow subgroup of order |D| of F ∗ (E) are SS-quasinormal in F ∗ (E), so Theorem 3.3 implies that F ∗ (E) is supersolvable. Hence F ∗ (E) = F (E) by Lemma 2.6. Let P be a Sylow p-subgroup of F (E), for some prime p, and let H be an arbitrary subgroup of order |D| of P . Since P is normal in G, it follows that H is subnormal in G. By the hypotheses and Lemma 2.2, H, not having a supersolvable supplement in G, is S-permutable in G. Thus, all subgroups of P of order |D| not having a supersolvable supplement in G are S-permutable in G. Applying Theorem 3.2, we conclude that G belongs to F. 4. PROOF OF THE MAIN THEOREMS ON S-QUASINORMALLY EMBEDDED SUBGROUPS Proof of Theorem 1.4. The sufficiency is obvious. We prove the necessity. Suppose the theorem is false and G is a counter-example of minimal order. We will derive a contradiction in several steps. Just as in the proof of Theorem 1.1, we have the following. (1) G has the unique minimal normal subgroup N such that G/N is p-nilpotent and Φ(G)=1. (2) Op (G)=1. (3) Op (G)=1. If Op (G) = 1, then N ≤ Op (G) by (1). Therefore, G has a maximal subgroup M such that G = M N and M ∩ N = 1. By (1), we have Φ(Op (G)) = 1. Thus, Op (G) ∩ M is normalized by N and M . Hence, by the uniqueness of N , we have N = Op (G). By Lemma 2.8, there is a maximal subgroup P1 of P such that P1 does not have p-nilpotent supplement. Since N has a p-nilpotent supplement M , we have N  P1 . Moreover, by the hypotheses, P1 is S-quasinormally embedded in G, then there is an S-quasinormal subgroup K of G such that P1 ∩ T ∈ Sylp (K). Assume that KG = 1, then N ≤ KG ≤ K. It follows that N ≤ P1 , and so P = N P1 = P1 ; a contradiction. So KG = 1. By Lemma 2.9, P1 is S-quasinormal in G. Thus, P1 ≤ N = Op (G), which is a contradiction. Thus, (3) holds. (4) A contradiction. Since G is a group of odd order, G is solvable, which contradicts (2) and (3). The proof of the theorem is complete. Proof of Theorem 1.5. It is clear that (i) implies (ii). Next, we prove that (ii) implies (i). Assume that the theorem is not true and let G be a counterexample of minimal order. We prove the theorem in the following several steps. Just as in the proof of Theorem 1.2, we have the following. (1) Op (G) = 1. (2) Op (G) = 1. (3) A contradiction. Since G is a group of odd order, G is solvable, which contradicts (1) and (2). The proof of the theorem is complete. By Sec. 3 and Theorem 1.5, we have the following result. Theorem 4.1. Let F be a saturated formation containing all supersolvable groups, and let G be a group of odd order with a normal subgroup E such that G/E ∈ F. Suppose that every noncyclic Sylow subgroup P of E has a subgroup D such that 1 < |D| < |P | and all subgroups H of P of order |H| = |D| not having a supersolvable supplement in G are S-quasinormally embedded in G. Then G ∈ F. By Theorem 1.3 in [20] and Lemma 2.2, we have the following results. MATHEMATICAL NOTES

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Theorem 4.2. Let F be a saturated formation containing all supersolvable groups, and let G be a group of odd order with a solvable normal subgroup E such that G/E ∈ F. Suppose that every noncyclic Sylow subgroup P of F (E) has a subgroup D such that 1 < |D| < |P | and all subgroups H of P of order |H| = |D| not having a supersolvable supplement in G are S-quasinormally embedded in G. Then G ∈ F. Theorem 4.3. Let G be a group of odd order with a normal subgroup E such that G/E is supersolvable. Suppose that every noncyclic Sylow subgroup P of F ∗ (E) has a subgroup D such that 1 < |D| < |P | and all subgroups H of P of order |H| = |D| not having a supersolvable supplement in G are S-quasinormally embedded in G. Then G is supersolvable. Proof of Theorem 1.6. The proof follows from Theorems 4.2 and 4.3. 5. APPLICATION Corollary 5.1. Let G be a group. If, for every prime p dividing the order of G and P ∈ Sylp (G), P has a subgroup D such that 1 < |D| < |P | and all subgroups H of P of order |H| = |D| and of order 2|D| (if P is a non-Abelian 2-group and |P : D| > 2) are SS-quasinormal in G, then G is supersolvable. Corollary 5.2. Let p be the smallest prime dividing the order of a group G, and let P be a Sylow p-subgroup of G. If P has a subgroup D such that 1 < |D| < |P | and all subgroups H of P of order |H| = |D| and of order 2|D| (if P is a non-Abelian 2-group and |P : D| > 2) are SS-quasinormal in G, then G is p-nilpotent. Corollary 5.3. Let G be a group. If, for every prime p dividing the order of G and all P ∈ Sylp (G), all maximal subgroups of P are SS-quasinormal in G, then G is supersolvable. Corollary 5.4. Let p be the smallest prime dividing the order of a group G, and let P be a Sylow p-subgroup of G. If all maximal subgroups of P are SS-quasinormal in G, then G is p-nilpotent. Corollary 5.5. Let G be a group. If all minimal subgroups or cyclic subgroups of order 4 are SS-quasinormal in G, then G is supersolvable. ACKNOWLEDGMENTS The authors are grateful to the referee and the editor who provided valuable suggestions and help. In particular, the authors are grateful to the referee, who provided a detailed report. The project is supported in part by the Chinese Universities Scientific Fund NSFC (grant no. 2014XJ015) and the CPSF (grant no. 2011M500168). REFERENCES 1. K. Doerk and T. Hawkes, Finite Soluble Groups, in de Gruyter Exp. Math. (Walter de Gruyter, Berlin, 1992), Vol. 4. 2. D. Gorenstein, Finite Groups, in Harper’s Ser. in Modern Math. (Harper & Row Publ., New York, 1968). 3. B. Huppert, Endliche Gruppen. I, in Grundlehren Math. Wiss. (Springer-Verlag, Berlin, 1967), Vol. 134. 4. L. A. Shemetkov, Formations of Finite Groups, in Contemporary Algebra (Nauka, Moscow, 1978) [in Russian]. 5. M. Assad and A. A. Heliel, “On S-quasinormally embedded subgroups of finite groups,” J. Pure Appl. Algebra 165 (2), 129–135 (2001). 6. A. Ballester-Bolinches and M. C. Pedraza-Aguilera, “Sufficient conditions for supersolubility of finite groups,” J. Pure Appl. Algebra 127 (2), 113–118 (1998). 7. W. Guo, A. N. Skiba, and K. P. Shum, “X-quasinormal subgroups,” Sibirsk. Mat. Zh. 48 (4), 742–759 (2007) [Siberian Math. J. 48 (4), 593–605 (2007)]. 8. W. Guo, K. P. Shum, and A. N. Skiba, “G-covering systems of subgroups for classes of p-supersolvable and p-nilpotent finite groups,” Sibirsk. Mat. Zh. 45 (3), 527–539 (2004) [Siberian Math. J. 45 (3), 433–442 (2004)]. MATHEMATICAL NOTES

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9. W. Guo, E. V. Legchekova, and A. N. Skiba, “Finite groups in which every 3-maximal subgroup commutes with all maximal subgroups,” Mat. Zametki 86 (3), 350–359 (2009) [Math. Notes 86 (3), 325–332 (2009)]. 10. O. H. Kegel, “Sylow-Gruppen und Subnormalteiler endlicher Gruppen,” Math. Z. 78, 205–221 (1962). 11. Shirong Li, Zhencai Shen, Jianjun Liu, and Xiaochun Liu, “The influence of SS-quasinormality of some subgroups on the structure of finite groups,” J. Algebra 319 (10), 4275–4287 (2008). 12. Shirong Li, Zhencai Shen, and Xianghong Kong, “On SS-quasinormal subgroups of finite groups,” Comm. Algebra 36 (12), 4436–4447 (2008). 13. L. Miao, “Finite groups with some maximal subgroups of Sylow subgroups M -supplemented,” Mat. Zametki 86 (5), 692–704 (2009) [Math. Notes 86 (5), 655–664 (2009)]. 14. Yangming Li and Yanming Wang, “On S-quasinormally embedded subgroups of finite group,” J. Algebra 281 (1), 109–123 (2004). 15. M. Ramadan, “The influence of S-quasinormality of some subgroups of prime power order on the structure of finite groups,” Arch. Math. (Basel) 77 (2), 143–148 (2001). 16. Zhencai Shen, Shirong Li, and Wujie Shi, “Finite groups with normally embedded subgroups,” J. Group Theory 13 (2), 257–265 (2010). 17. Zhencai Shen, Wujie Shi, and Qingliang Zhang, “S-quasinormality of finite groups,” Front. Math. China 5 (2), 329–339 (2010). 18. Zhencai Shen and Wujie Shi, “The influence of SNS-permutability of some subgroups on the structure of finite groups,” Publ. Math. Debrecen 78 (1), 159–168 (2011). 19. L. A. Shemetkov and A. N. Skiba, “On the X Φ-hypercentre of finite groups,” J. Algebra 322 (6), 2106–2117 (2009). 20. A. N. Skiba, “On weakly s-permutable subgroups of finite groups,” J. Algebra 315 (1), 192–209 (2007). 21. A. N. Skiba and O. V. Titov, “Finite groups with C-quasinormal subgroups,” Sibirsk. Mat. Zh. 48 (3), 674– 688 (2007) [Siberian Math. J. 48 (3), 544–554 (2007)]. 22. A. N. Skiba, “A note on c-normal subgroups of finite groups,” Algebra Discrete Math., No. 3, 85–95 (2005). 23. Xiaolan Yi and L. A. Shemetkov, “The formation of finite groups with a supersolvable π-Hall subgroup,” Mat. Zametki 87 (2), 280–286 (2010) [Math. Notes 87 (2), 258–263 (2010)].

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