Monatsh Math (2011) 162:329–339 DOI 10.1007/s00605-009-0175-2

On finite groups with prime-power order S-quasinormally embedded subgroups Xianbiao Wei · Xiuyun Guo

Received: 9 July 2009 / Accepted: 11 November 2009 / Published online: 28 November 2009 © Springer-Verlag 2009

Abstract A subgroup H of a finite group G is said to be S-quasinormally embedded in G if for each prime p dividing the order of H , a Sylow p-subgroup of H is also a Sylow p-subgroup of some S-quasinormal subgroup of G. In this paper we investigate the structure of finite groups that have some S-quasinormally embedded subgroups of prime-power order, and new criteria for p-nilpotency are obtained. Keywords S-Quasinormal subgroups · S-Quasinormally embedded subgroups · p-Nilpotent groups Mathematics Subject Classification (2000)

20D10 · 20D20

1 Introduction All groups considered in this paper are finite. Two subgroups H and K of a group G are said to be permutable if H K = K H . It is clear that H and K are permutable

Communicated by J.S. Wilson. The research of the work was partially supported by the National Natural Science Foundation of China (10771132), SRFDP (200802800011), the Research Grant of Shanghai University, Shanghai Leading Academic Discipline Project (J50101) and NSF of Anhui Provence (KJ2008A030). X. Wei · X. Guo (B) Department of Mathematics, Shanghai University, 200444 Shanghai, People’s Republic of China e-mail: [email protected]; [email protected] X. Wei Department of Mathematics and Physics, Anhui Institute of Architecture and Industry, 230022 Hefei, People’s Republic of China e-mail: [email protected]

123

330

X. Wei, X. Guo

if and only if H K is a subgroup of G. A subgroup H of a group G is said to be S-quasinormal in G if H permutes with every Sylow subgroup of G. This concept was introduced by Kegel in [9], and it has been investigated extensively by many authors; for example, see [1,2,4,17]. Recently, Ballester-Bolinches and Pedraza-Aguilera in [5] introduced the concept of an S-quasinormally embedded subgroup. A subgroup H of a group G is said to be S-quasinormally embedded in G if for each prime p dividing the order of H , a Sylow p-subgroup of H is also a Sylow p-subgroup of some S-quasinormal subgroup of G. It is clear that every S-quasinormal subgroup is a S-quasinormally embedded subgroup. However, the converse is not true in general because every Sylow subgroup of a group G is a S-quasinormally embedded subgroup. Since this concept was introduced, many authors have investigated the influence of S-quasinormally embedded subgroups on the structure of groups (see [3,12,13]). For instance, Asaad and Heliel in [3] proved that a group G is p-nilpotent if and only if every maximal subgroup of P is S-quasinormally embedded in G, where p is the smallest prime dividing the order of G and P is a Sylow p-subgroup of G. Later on, Li and Wang in [12] showed that a group G is p-nilpotent if N G (P) is p-nilpotent and every maximal subgroup of a Sylow p-subgroup P of G is S-quasinormally embedded in G. In this paper, we study the structure of groups in which every subgroup of order p m is S-quasinormally embedded in the group, where p is a prime and m is a fixed positive integer. Some necessary and sufficient conditions related to p-nilpotency are obtained. For example, we prove the following results: Theorem 3.7 Let p be the smallest prime dividing the order of a group G and P a Sylow p-subgroup of G. Then G is p-nilpotent if and only if there is a subgroup D of P with 1 < |D| < |P| such that every subgroup of P with order |D| or 4 (if |D| = 2) is S-quasinormally embedded in G. Theorem 3.9 Let p be a odd prime dividing the order of a group G and P a Sylow p-subgroup of G. Then G is p-nilpotent if and only if NG (P) is p-nilpotent and there is a subgroup D of P with 1 < |D| < |P| such that every subgroup of P with order |D| is S-quasinormally embedded in G. 2 Preliminaries In this section, we list some lemmas which will be useful for the proofs of our main results. Lemma 2.1 [5, Lemma 1] Suppose that U is S-quasinormally embedded in a group G, and let H ≤ G and K  G. Then the following assertions hold. (a) If U ≤ H , then U is S-quasinormally embedded in H . (b) U K is S-quasinormally embedded in G and U K /K is S-quasinormally embedded in G/K . (c) If K ≤ H and H/K is S-quasinormally embedded in G/K , then H is S-quasinormally embedded in G. Lemma 2.2 [11, Lemma 2.2] Let H be a nilpotent subgroup of a group G. Then the following statements are equivalent:

123

On finite groups with prime-power order S-quasinormally embedded subgroups

331

(1) H is S-quasinormal in G; (2) H ≤ F(G) and H is S-quasinormally embedded in G. Lemma 2.3 [9] An S-quasinormal subgroup of a group G is subnormal in G. Lemma 2.4 [6, Theorem 1] If a subgroup H of a group G is S-quasinormal in G, then H/HG is nilpotent. Lemma 2.5 [15, Lemma A] If H is S-quasinormal in a group G and H is a p-group for some prime p, then O p (G) ≤ N G (H ). Lemma 2.6 [10, Lemma 2.6] Let N be a nontrivial solvable normal subgroup of a group G. If N ∩ (G) = 1 then the Fitting subgroup F(N ) of N is the direct product of some minimal normal subgroups of G. Lemma 2.7 Let N be an elementary abelian normal p-subgroup of a group G. If there is a subgroup D of P with 1 < |D| < |N | such that every subgroup of N with order |D| is S-quasinormally embedded in G, then there exists a maximal subgroup M of N such that M is normal in G. Proof Let N1 be a maximal subgroup of N . By Lemma 2.2, every subgroup of N with order |D| is S-quasinormal in G and therefore N1 is S-quasinormal in G since N1 is a product of some subgroups of N of order |D|. It follows from Lemma 2.5 that |G : N G (N1 )| = p α1 for some integer α1 , and so p divides the number of maximal subgroups of N . Thus, by [8, III.8.5], there is a maximal subgroup of N which is normal in G. Lemma 2.8 [16, Theorem 1] Let A be a p  -group of automorphisms of a p-group P, with p an odd prime. If every subgroup of P with prime order is A-invariant, then A is cyclic. 3 Main results In this section, we give some necessary and sufficient conditions for a group to be p-nilpotent. First we prove some lemmas. Lemma 3.1 Let P be a Sylow p-subgroup of a group G with O p (G) = 1 and let H be a normal subgroup of P. If every proper subgroup of G containing P is p-nilpotent and H is S-quasinormally embedded in G, then either H is normal in G or there exists a subnormal subgroup T of G such that H is a Sylow p-subgroup of T and [G : T ] = [P : H ]. Proof By our hypotheses, there exists a subgroup T of G such that H is a Sylow p-subgroup of T and T is S-quasinormal in G. By Lemma 2.3, T is subnormal in G. On the other hand, if TG = 1, then by Lemma 2.4, T = T /TG is nilpotent. It follows that T ≤ F(G) = O p (G) since O p (G) = 1. Consequently, H = T is S-quasinormal in G. By Lemma 2.5, O p (G) ≤ N G (H ). Hence H  P N G (H ) = G, in contradiction to TG = 1. Hence TG > 1. If P TG < G, then by the hypotheses

123

332

X. Wei, X. Guo

P TG is p-nilpotent and therefore TG is a p-group. Since T /TG ≤ F(G/TG ) we have H/TG ≤ O p (G/TG ) = O p (G)/TG . Lemma 2.2 implies that H is S-quasinormal in G. By the same argument as above, H is normal in G, as desired. If P TG = G, then it is easy to see that T = H TG and [G : T ] = [P : H ]. Lemma 3.2 Let H be a cyclic subgroup of G of order 4. If O2 (G) = 1 and H is S-quasinormally embedded in G, then H is S-quasinormal in G. Proof By our hypotheses, there exists a subgroup K of G such that H is a Sylow subgroup of K and K is S-quasinormal in G. Then K is 2-nilpotent by Burnside’s theorem [14, Theorem 10.1.8]. Let K 2 be a normal Hall 2 -subgroup of K and L a minimal normal subgroup of K 2 . Then the Feit–Thompson odd-order theorem implies that L is an abelian group of prime-power order. It follows that L is a subnormal subgroup of G and so L ≤ F(G) = O2 (G), a contradiction. Thus K 2 = 1 and hence H = K is S-quasinormal in G. Lemma 3.3 Let P be a 2-subgroup of a group G and N a normal subgroup of G of order 2 contained in P. If O2 (G) = 1 and every subgroup of order 4 of P is S-quasinormally embedded in G, then every minimal subgroup of P is S-quasinormal in G. Proof Let K be a subgroup of P of order 2 with K = N . Then A = K N is an elementary abelian 2-group of order 4. Thus there exists a subgroup T of G such that A is a Sylow 2-subgroup of T and T is S-quasinormal in G. By Burnside’s theorem, we may assume that B/N is a normal Hall 2 -subgroup of T /N . Since |N | = 2 we have B = N × B2 . If B2 = 1, then there exists a normal abelian subgroup C of B2 with prime-power order since B2 is solvable by the odd-order theorem. Thus C is subnormal in G as C  B2  B  T and T is subnormal in G by Lemma 2.3. Furthermore, C ≤ F(G) = O2 (G), a contradiction. So T = A is S-quasinormal in G and therefore A ≤ O2 (G). Let Q be a Sylow q-subgroup of G with q = 2. Since N Q  AQ and N Q is 2-nilpotent, we see that K normalizes Q and therefore K Q is a subgroup of G. Thus K is S-quasinormal in G. If P is a p-group for some prime p we shall write M d(P) = {P1 , P2 , . . . , Pd } to Pi = (P) and d is as denote any set of maximal subgroups of P such that i=1 small as possible. Lemma 3.4 Let p be a prime dividing the order of a group G and let P be a Sylow p-subgroup of G. If N G (P) is p-nilpotent and every element in M (P) is S-quasinormally embedded in G, then G is p-nilpotent. Proof Let M (P) = {P1 , P2 , . . . , Pd }. If O p (G) = 1, then every element of M (P) = {P1 , P2 , . . . , Pd } is S-quasinormally embedded in G = G/O p (G) by Lemma 2.1(b), where Pi = Pi O p (G)/O p (G). Since N G (P) = N G (P) is p-nilpotent, by induction, G is p-nilpotent and therefore G is p-nilpotent. Now we may assume that O p (G) = 1.

123

On finite groups with prime-power order S-quasinormally embedded subgroups

333

Let H be an element in M (P). Then by Lemma 3.1, either H is normal in G or there exists a subnormal subgroup T of G such that H is a Sylow p-subgroup of T and [G : T ] = [P : H ]. If H is normal in G, then the p-nilpotency of N G (P) implies that N G/H (P/H ) = C G/H (P/H ) and therefore G/H is p-nilpotent. Then there exists a Sylow q-subgroup Q of G such that P Q is a subgroup of G for any q ∈ π(G) with q = p by [7, Theorem 6.3.5]. If P Q < G, then P Q is p-nilpotent by induction. Hence Q ≤ C G (O p (G)) ≤ O p (G) by [7, Theorem 6.3.2], a contradiction. Thus G = P Q is solvable. Let N be a minimal normal subgroup of G contained in H . If N ≤ Pi ∈ M (P) for all i, then N ≤ (P). It is easy to see that G/N satisfies the hypotheses of the theorem. So G/N is p-nilpotent by induction. It follows from [14, Theorem 5.2.13] that G/(G) is p-nilpotent and so G is p-nilpotent, as desired. Thus we may assume that N  (P) and therefore there that is an element in M (P), P1 say, such that P = N P1 . By the hypotheses, there exists a subgroup T1 of G such that P1 is a Sylow p-subgroup of T1 and T1 is S-quasinormal in G. If N T1 < G, then N T1 < G is p-nilpotent by induction and hence T1 = P1 is a p-group. It follows from Lemma 2.5 that P1 is normal in G and so P = N P1 is normal in G. Hence G = N G (P) is p-nilpotent, as desired. So we may assume that N T1 = G. The minimality of N implies that N ∩ T1 = 1 and so N ∩ P1 = 1. Thus |N | = |N : N ∩ T1 | = |P : P1 | = p. Now we may assume that H ∩ (P) = 1 by using the Lemma 2.6 and therefore that (P) = 1 since P/H has order p. The p-nilpotency of N G (P) implies that N G (P) = C G (P) and so G is p-nilpotent by Burnside’s theorem, as desired. Now we may assume that G has a subnormal subgroup T such that H ∈ Syl p (T ) and [G : T ] = [P : H ] = p. Then T is a maximal subgroup of G and therefore T is normal in G. By the arbitrariness of H , for every element Pi in M (P), there exists ofTi . Let a normal d d subgroup Ti such that G = P Ti and Pi is a Sylow p-subgroup Ti . Then K  G and G/K is a p-group. Since P ∩ K = i=1 (P Ti ) = K = i=1 d i=1 Pi = (P), K is p-nilpotent by Tate’s theorem [8, IV.4.7]. This implies that G is p-nilpotent. The proof of the lemma is complete. Lemma 3.5 Let p be an odd prime dividing the order of a group G and P a Sylow p-subgroup of G. If N G (P) is p-nilpotent and every minimal subgroup of P is S-quasinormally embedded in G, then G is p-nilpotent. Proof If O p (G) = 1, then it is easy to see that G/O p (G) satisfies the hypotheses of the lemma. By induction, G/O p (G) is p-nilpotent and hence G is p-nilpotent. Now we may assume that O p (G) = 1. Let J (P) be the Thompson subgroup of P. Then N G (P) ≤ N G (Z (J (P))) and every minimal subgroup of P is S-quasinormally embedded in N G (Z (J (P))) by Lemma 2.1(a). If N G (Z (J (P))) < G, then, by induction, N G (Z (J (P))) is p-nilpotent. It follows from [7, Theorem 8.3.1] that G is p-nilpotent. Thus we may assume that N G (Z (J (P))) = G and hence P > O p (G) = 1. By [7, Theorem 8.3.1] again, G/O p (G) is p-nilpotent and therefore G is p-solvable. Then there exists a Sylow q-subgroup Q of G such that P Q is a subgroup of G for any q ∈ π(G) with q = p by [7, Theorem 6.3.5]. If P Q < G, then P Q is p-nilpotent by induction. It follows from [14, Theorem 9.3.1] that Q ≤ C G (O p (G)) ≤ O p (G), a contradiction. Hence we may assume that G = P Q. Let U = 1 (O p (G)) and

123

334

X. Wei, X. Guo

choose x ∈ U with order p. Then, by Lemma 2.2, X = x is S-quasinormal in G and therefore Q normalizes X . If Q = C Q (X ) for every X , then Q centralizes U . By [8, IV.5.12], Q centralizes O p (G) and therefore Q ≤ C G (O p (G)) ≤ O p (G), a contradiction. So there exists an element x ∈ U such that Q > C Q (X ). Since Q/C Q (X )  Aut(X ), q divides p − 1, the order of Aut(X ), and hence q < p. On the other hand, if C Q (U ) > 1, then, by [8, IV.5.12] again, C Q (U ) centralizes O p (G). Moreover, C Q (U ) ≤ C G (O p (G)) ≤ O p (G), a contradiction. Hence C Q (U ) = 1 and so Q = Q/C Q (U )  Aut(U ). By Lemma 2.8, Q is cyclic and hence G is q-nilpotent by Burnside’s theorem [14, 10.1.8]. It follows that N G (P) = G is p-nilpotent. Remark 3.6 The assumption that N G (P) is p-nilpotent in Lemma 3.5 is essential. For example, a Sylow 3-subgroup of the symmetric group S4 is S-quasinormally embedded in S4 but S4 is not 3-nilpotent. Now we prove our main results. Theorem 3.7 Let p be the smallest prime dividing the order of a group G and P a Sylow p-subgroup of G. Then G is p-nilpotent if and only if there is a subgroup D of P with 1 < |D| < |P| such that every subgroup of P with order |D| or 4 (if |D| = 2) is S-quasinormally embedded in G. Proof Let G = P  O p (G) and let H be a maximal subgroup of P. Since |G : H O p (G)| = p, H O p (G) is normal in G as p is the smallest prime dividing the order of the group G. Thus H is S-quasinormally embedded in G. Conversely, suppose the result is false and let G be a counterexample of minimal order. Then (1) O p (G) = 1. If O p (G) = 1, then G = G/O p (G) satisfies the hypotheses of our theorem by Lemma 2.1(b). The choice of G implies that G is p-nilpotent and therefore G is p-nilpotent, a contradiction. (2) If L is a proper subgroup of G with |L| p > |D|, then L is p-nilpotent. Let L p be a Sylow p-subgroup of L and H a subgroup of L p with |H | = |D|. By Sylow’s theorem, H < L p ≤ P x for some x ∈ G. Then H is S-quasinormally −1 embedded in G since H x is S-quasinormally embedded in G. Lemma 2.1(a) implies that H is S-quasinormally embedded in L. If |D| = 2, then the above methods also show that every subgroup of L p with order 4 is S-quasinormally embedded in L. The choice of G implies that L is p-nilpotent. (3) |P : D| > p. This follows from [3, Theorem 3.1]. (4) |D| > p. Suppose |D| = p. Let K be a proper subgroup of G. If p < |K | p , then K is p-nilpotent by (2). If |K | p = p, then K is also p-nilpotent since p is the smallest prime dividing the order of G. Thus G is a minimal subgroup with respect to being p-nilpotent (that is, its proper subgroups are all p-nilpotent but it itself is not p-nilpotent). By [8, IV.5.4], G = P  Q where P is a Sylow p-subgroup of G and Q a Sylow q-subgroup of G with p = q. Then it follows from Lemma 2.2

123

On finite groups with prime-power order S-quasinormally embedded subgroups

335

that every subgroup of P of order p or 4 (if |D| = 2) is S-quasinormal in G. By [17, Theorem 2], G is p-nilpotent, a contradiction. (5) O p (G) = 1. Let H be a subgroup of G of order |D|. Then there exists a subgroup K of G such that H is a Sylow p-subgroup of K and K is S-quasinormal in G. If K G = 1, then H is a nilpotent subnormal subgroup of G and so H = K ≤ F(G) = O p (G) by (1). Now we assume that K G = 1. Let P1 be a maximal subgroup of P such that (K G ) p ≤ P1 where (K G ) p is a Sylow p-subgroup of K G . By (3) and (2), P1 K G is p-nilpotent and so is K G . By (1), K G is a p-group. In the rest of the proof, we always assume that N is a normal subgroup of G contained in P with minimal possible order. (6) G/N is p-nilpotent. By Lemma 2.7, |D| ≥ |N |. Now we distinguish the following cases. Case 1. Suppose that |D| > |N |. If p > 2 or p = 2 and |D| > 2|N |, then it is easy to see that G/N satisfies the hypotheses of the theorem. The choice of G implies that G/N is p-nilpotent. Now we may assume that p = 2 and |D| = 2|N |. If N is cyclic, then |N | = 2 and |D| = 4. Thus every subgroup of P with order 2 or 4 is S-quasinormal in G by Lemmas 3.2 and 3.3, contrary to (4). Thus N is not cyclic. Let K /N be a subgroup of P/N of order 2. By Lemma 2.1(b), K /N is S-quasinormally embedded in G/N as K is S-quasinormally embedded in G. Now let H/N be a subgroup of P/N of order 4 and H1 a maximal subgroup of H . We will prove that H/N is S-quasinormally embedded in G/N . If N  H1 , then H/N = H1 N /N is S-quasinormally embedded in G/N by Lemma 2.1(b) as H1 is S-quasinormally embedded in G. Thus we may assume that N ≤ (H ) and therefore N ≤ (G) by [14, Theorem 5.2.11]. Then there exists a subgroup T of G such that H1 is a Sylow 2-subgroup of T and T is S-quasinormal in G. Since N ≤ T, N ≤ TG . By (3) and (2), H TG is 2-nilpotent and so is TG . By (1), TG is a 2-group. Notice that N ≤ TG ≤ H1 < H and |H/N | = 4, we have that either N = H1 or N = TG . If TG = H1 , then there exists a maximal subgroup H2 of H such that H = H1 H2 . By Lemma 2.1(b) and (c), H/N is S-quasinormally embedded in G/N , as desired. If TG = N , then T is nilpotent since T /TG is nilpotent and N ≤ (G). Therefore T ≤ F(G) = O p (G) by (1). Then T is a 2-group and therefore H1 = T is S-quasinormal in G. Now we have proved that every maximal subgroup of H is S-quasinormal in G. Since H is not cyclic, there exist maximal subgroups H2 and H3 such that H = H2 H3 . Hence H/N = H2 H3 /N is S-quasinormal in G/N ; in particular, H/N is S-quasinormally embedded in G/N . Case 2. Suppose that |D| = |N |. By (4), |N | > p and hence N is non-cyclic. Let K /N be a subgroup of P/N with order p. Since K is non-cyclic, there exists a maximal subgroup K 1 of K such that K = K 1 N . By Lemma 2.1(b), K /N is S-quasinormally embedded in G/N .

123

336

X. Wei, X. Guo

Now let p = 2, H/N a subgroup of P/N of order 4 and A a maximal subgroup of H such that N is a maximal subgroup of A. Then there exists a maximal subgroup A1 of A such that A = N A1 . Since A1 is S-quasinormally embedded in G, so is A by Lemma 2.1(b). Consequently, there is a subgroup B such that A is a Sylow subgroup of B and B is S-quasinormal in G. Since |B| p > |D|, B is p-nilpotent by (2). If B is not a p-group and let B p be the normal p-complement, then by the odd-order theorem there is a normal abelian subgroup M of B p such that M ≤ F(G) = O p (G), a contradiction. So B = A is S-quasinormal in G. Thus we have proved that every maximal subgroup of H containing N is S-quasinormal in G. If H/N is not cyclic, then there are maximal subgroups H1 /N and H2 /N of H/N such that H/N = (H1 /N )(H2 /N ). Since both H1 and H2 are S-quasinormal in G by the above argument, H/N is S-quasinormal in G/N . If H/N is cyclic, then N  (H ) and therefore there is a maximal subgroup H3 of H such that H = H3 N . Let H4 be a maximal subgroup of H such that N < H4 < H . If H3 is cyclic and let X be the maximal subgroup of H3 , then H4 = X N . By the hypotheses, there is a subgroup T such that X is a Sylow subgroup of T and T is S-quasinormal in G. Then it follows that T is 2-nilpotent. Let T = X  T2 . If T2 = 1, then there is a normal abelian subgroup C of T2 . Since C  T2  T and T is subnormal in G, it follows that C ≤ F(G) = O2 (G), a contradiction. So T = X is S-quasinormal in G and therefore H3 O 2 (G) ≤ N G (X ). If N G (X ) < G, then, by (2), N G (X ) is 2-nilpotent and so is O 2 (G), contrary to (1). Thus X is normal in G and hence there is a normal subgroup of G of order 2, which contradicts |N | = |D| > 2 by (4). Now we may assume that H3 is non-cyclic. Then there is a maximal subgroup S of H3 such that S N = H4 . If S N is a maximal subgroup of H , then S N is S-quasinormal in G by the above argument and hence H/N = (S N )H4 /N is S-quasinormal in G/N . If S N = H , then, by Lemma 2.1(b), H/N is S-quasinormally embedded in G/N . From Cases 1 and 2, we see that every subgroup of P/N of order p or 4 (if p = 2) is S-quasinormally embedded in G/N . By the choice of G, G/N is p-nilpotent. (7) Conclusion. By (6), there exists a normal subgroup M of G such that |G : M| = p. Since |M| p > |D| by (3), it follows from (2) that M is p-nilpotent. Let M p be the normal p-complement in M. By (1), M p = 1 and therefore M is a p-group, the final contradiction. Thus the proof of the theorem is complete. Corollary 3.8 Let G be a group and suppose that for every prime p dividing |G| a Sylow p-subgroup P has a non-trivial proper subgroup D such that every subgroup of P of order |D| or 4 (when |D| = 2) is S-quasinormally embedded in G. Then G has a Sylow tower of supersolvable type. Theorem 3.9 Let p be an odd prime dividing the order of a group G and P a Sylow p-subgroup of G. Then G is p-nilpotent if and only if NG (P) is p-nilpotent and there is a subgroup D of P with 1 < |D| < |P| such that every subgroup of P with order |D| is S-quasinormally embedded in G. Proof If G is p-nilpotent, then G = P  O p (G). Let H be a maximal subgroup of P. Since H O p (G)  P, O p (G) = G, H is S-quasinormally embedded in G.

123

On finite groups with prime-power order S-quasinormally embedded subgroups

337

Conversely, suppose the theorem is false and let G be a counterexample with minimal order. Then (1) O p (G) = 1. Suppose that O p (G) = 1. It is easy to see that N G (P) = N G (P) is p-nilpotent and every subgroup of P with order |D| is S-quasinormally embedded in G = G/O p (G) by Lemma 2.1(b). Thus G satisfies the hypotheses of the theorem. The choice of G implies that G is p-nilpotent and so G is p-nilpotent, a contradiction. (2) If L is a proper subgroup of G with P ≤ L, then L is p-nilpotent. In fact, N L (P) is p-nilpotent and every subgroup of P of order |D| is S-quasinormally embedded in L by Lemma 2.1(a), and the minimality of G implies that L is p-nilpotent. (3) O p (G) = 1 and G = P Q where Q ∈ Sylq (G) and q = p. Let J (P) be the Thompson subgroup of P. Then N G (P) ≤ N G (Z (J (P))) ≤ G and every subgroup of P of order |D| is S-quasinormally embedded in N G (Z (J (P))) by Lemma 2.1(a). If N G (Z (J (P))) < G, then, by (2), N G (Z (J (P))) is p-nilpotent and therefore G is p-nilpotent by [7, Theorem 8.3.1], a contradiction. So Z (J (P))  G, and this leads to 1 = O p (G) < P. Consider G = G/O p (G) and let G 1 be the inverse imagine of N G (Z (J (P))) in G. Since O p (G) is the largest normal subgroup of G contained in P, we have N G (P) ≤ G 1 < G. By (2), G 1 is p-nilpotent and by [7, Theorem 8.3.1] again, G is p-nilpotent. Then there exists a Sylow q-subgroup Q of G such that P Q is a subgroup of G for any q ∈ π(G) with q = p by [7, Theorem 6.3.5]. If P Q < G, then P Q is p-nilpotent by (2). Hence Q ≤ C G (O p (G)) ≤ O p (G) by [7, Theorem 6.3.2], a contradiction. Thus P Q = G. Now we always assume that N is a minimal normal subgroup of G contained in P. (4) |N | ≤ |D|. If |N | > |D|, then by Lemma 2.7 there is a maximal subgroup of N which is normal in G, contrary to the minimality of N . (5) G/N is p-nilpotent and N = F(G) = O p (G). If |N | < |D|, it is easy to see that G/N satisfies the hypotheses of our theorem and therefore G/N is p-nilpotent by the choice of G. Now we assume that |N | = |D|. Then |D| > p by Lemma 3.5 and hence N is non-cyclic. Let H/N be a subgroup of P/N of order p. Then there is a maximal subgroup L of H such that H = L N . Since L is S-quasinormally embedded in G, H/N is S-quasinormally embedded in G/N by Lemma 2.2. Since N G/N (P/N ) = N G (P)/N is p-nilpotent, we see that G/N satisfies the hypotheses of the theorem. The choice of G implies that G/N is p-nilpotent. Since the class of all p-nilpotent groups is a saturated formation, N is the unique minimal normal subgroup of G contained in P. By (1) and (3), N is the unique minimal normal subgroup of G and therefore (G) = 1. Further, we have N = F(G) = O p (G). (6) Conclusion. Since G is solvable by (3), there is a maximal subgroup M of G such that |G : M| is a prime. If |G : M| = p, then M is p-nilpotent by (2) and therefore P = M G

123

338

X. Wei, X. Guo

by (1), a contradiction. Thus we may assume that |G : M| = p. Then it follows that P ∩ M is a maximal subgroup of P and also a Sylow p-subgroup of M. If N G (P ∩ M) < G, then N G (P ∩ M) is p-nilpotent by (2) and so is N M (P ∩ M). Since |P : D| > p by Lemma 3.4, every subgroup of P ∩ M of order |D| is S-quasinormally embedded in M by Lemma 2.1(a). Consequently, M satisfies the hypotheses of our theorem and therefore the choice of G implies that M is p-nilpotent, a contradiction. Hence P ∩ M  G and N = O p (G) = P ∩ M is a maximal subgroup of P by (5). This leads to |D| < |N | by Lemma 3.4, in contradiction to (4), the final contradiction. Thus the proof of the theorem is complete. Corollary 3.10 Let N be a normal subgroup of a group G and let p be an odd prime dividing |N | such that G/N is p-nilpotent. If N G (P) is p-nilpotent where P ∈ Syl p (N ) and if P has a subgroup D with 1 < |D| < |P| such that every subgroup of P of order |D| is S-quasinormally embedded in G, then G is p-nilpotent. Proof It is clear that N N (P) is p-nilpotent and every subgroup H of P of order |D| is S-quasinormally embedded in N . By Theorem 3.9, N is p-nilpotent. Let N p be the Hall p  -subgroup of N ; then N p is normal in G. If N p = 1, we consider G/N p . By Lemma 2.1, every subgroup H N p /N p of P N p /N p with |H N p /N p | = |D| is S-quasinormally embedded in G/N p . Since (G/N p )/(N /N p ) G/N is p-nilpotent, G/N p is p-nilpotent by induction and so G is p-nilpotent, as desired. Thus we may assume that N p = 1 and N = P. By the hypotheses of the corollary, N G (P) = G is p-nilpotent. Acknowledgments The authors would like to thank the referee for valuable suggestions and comments that contributed to the final version of this paper. The authors also would like to thank Professor John Wilson for helping us to polish the final version of this paper.

References 1. Asaad, M.: On maximal subgroups of Sylow subgroups of finite groups. Comm. Algebra 26, 3647– 3652 (1998) 2. Asaad, M., Csögö, P.: Characterization of finite groups with some S-quasinormal subgroups. Monatsh. Math. 146, 263–266 (2005) 3. Asaad, M., Heliel, A.A.: On S-quasinormally embedded subgroups of finite groups. J. Pure Appl. Algebra 165, 129–135 (2001) 4. Asaad, M., Ramadam, M., Shaalan, A.: Influence of π -quasinormality on maximal subgroups of Sylow subgroups of fitting subgroups of a finite group. Arch. Math. (Basel) 51, 521–527 (1991) 5. Ballester-Bolinches, A., Pedraza-Aguilera, M.C.: Sufficient conditions for supersolvability of finite groups. J. Pure Appl. Algebra 127, 113–118 (1998) 6. Deskins, W.E.: On quasinormal subgroups of finite groups. Math. Z. 82, 125–132 (1963) 7. Gorenstein, D.: Finite Groups. Harper and Row, New York (1968) 8. Huppert, B.: Endliche Gruppen I. Springer, New York (1967) 9. Kegel, O.: Sylow–Gruppen und Subnormalteiler endlicher Gruppen. Math. Z. 78, 205–221 (1962) 10. Li, D., Guo, X.: The influence of c-normality of subgroups on the structure of finite groups. J. Pure Appl. Algebra 150, 53–60 (2000) 11. Li, S., Shen, Z., Liu, J., Liu, X.: The influence of SS-quasinormality of some subgroups on the structure of finite groups. J. Algebra 319, 4275–4287 (2008) 12. Li, Y., Wang, Y.: On π -quasinormally embedded subgroups of finite groups. J. Algebra 281, 109–123 (2004)

123

On finite groups with prime-power order S-quasinormally embedded subgroups

339

13. Li, Y., Wang, Y., Wei, H.: On p-nilpotency of finite grroups with some subgroups π -quasinormally embeeded. Acta Math. Hungar. 108, 283–298 (2005) 14. Robinson, D.J.S.: A Course in the Theory of Groups. Springer, New York (1982) 15. Schmid, P.: Subgroups permutable with all Sylow subgroups. J. Algebra 207, 285–293 (1998) 16. Sergienko, V.I.: A criterion for the p-solubility of finite groups. Mat. Zametki 9, 216–220 (1971) 17. Shaalan, A.: The Influence of π -quasinormality of some subgroups on the structure of a finite group. Acta Math. Hung. 56, 287–293 (1990)

123