Basic notions and notations
G is everywhere a finite group.
π(G): the set of prime divisors of the order ofG.
Φ(G): the Frattini subgroup ofGis the intersection of all maximal subgroups ofG.
F(G): the Fitting subgroup of G is the unique maximal nilpotent normal subgroup ofG.
F∗(G): the generalized Fitting subgroup ofGis the set of all elements x ofG which induce an inner automorphism on every chief factor ofG.
Definition: H is a Hall subgroup of G, if (|H|,|G:H|) = 1.
Definition: We say thatH is a subnormal subgroup of G, if there exists H =H0⊳ H1⊳ . . . ⊳ Hk =G.
We denote it byH ⊳ ⊳ G.
Definition: Gis aT-group if every its subnormal subgroup is normal inG.
Definition: His anS-quasinormal subgroup ofG, if it permutes with every Sylow subgroup ofG (i.e.HQ=QH for all Sylow subgroups Qof G).
Definition: G is a T∗-group, if every its subnormal subgroup is S-quasinormal inG.
Definition: K is anH-subgroup ofG, ifNG(K)∩Kg ⊆K for everyg∈G.
Definition: Gis supersolvable if every its chief factor is cyclic.
Definition: Gisp-nilpotent ifGhas a normalp-complementN i.e.N ⊳G, G=N P, N∩P = 1 withP ∈Sylp(G).
Definition: A classF of groups is called aformationifFcontains all homomorphic images of a group inF and ifG/M andG/N are inF, thenG/M∩N is inF for normal subgroupsM, N ofG. Aformation F is said to be saturated, if G/Φ(G)∈ F impliesG∈ F.
2.1. The influence of minimal subgroups on the structure of finite groups
Minimal subgroups and S-quasinormality
Many authors have investigated the structure of a finite groupGunder the assump-tion that certain minimal subgroups of G are well situated in G. Ito [Hu1, p. 283]
showed that a group of odd order is nilpotent provided that all minimal subgroups ofGlie in the center of the group. Buckley [Bu] proved that a groupGof odd order is supersolvable (i.e. its every chief factor is cyclic) if all minimal subgroups ofGare normal.
Clearly if H is a normal subgroup of a group G, then HK = KH for every subgroup K of G, i.e. H permutes with every subgroup of G. A subgroup of G is called S-quasinormal (orπ-quasinormal) if it permutes with all Sylow subgroups of G. Thus S-quasinormality can be considered as a weak form of normality. The concept ofS-quasinormality (π-quasinormality) was introduced by Kegel [Ke].
Then many authors studied the influence ofS-quasinormality of some subgroups which ensures the supersolvability of the group. Yokoyama [Yo1, Yo2] and Laue [La] extended the results of Ito and Buckley, using formation theory to generalize to notion of centrality.
Recall that a classFof groups is called aformationifFcontains all homomorphic images of a group inF and ifG/M andG/N are inF, thenG/(M∩N) is inF for normal subgroupsM, N ofG. AformationF is said to besaturated, ifG/Φ(G)∈ F impliesG∈ F (see [Hu1, p. 696]). Throughout this chapter U will denote the class of supersolvable groups. ClearlyU is a formation. Since a groupGis supersolvable iffG/Φ(G) is supersolvable [Hu1, VI. p. 173] it follows U is a saturated formation.
M. Asaad, A. Ballester-Bolinches and M. C. Pedraza Aguilera in [AsBP] proved the following statement:
Let F be a saturated formation, containing the class U of supersolvable groups.
Suppose that G is a group with a normal subgroup H such that G/H∈ F. If every subgroup of H of prime order or order 4 isS-quasinormal in G, then G∈ F.
It is natural to limit the hypotheses on minimal subgroups to a smaller sub-group. So with M. Asaad in [ACs1] we generalized the above-mentioned result in the following way:
Theorem 2.1.1[ACs1, Theorem 1]. LetF be a saturated formation containing the class U and let G be a group. Equivalent are:
(a) G∈ F.
(b) There is a normal solvable subgroup H in G such that G/H ∈ F and the subgroups of prime order or order4of the Fitting subgroupF(H)areS-quasinormal in G.
Immediate corollaries of this theorem are:
Corollary 2.1.2[ACs1, Corollary 1, 2]. i)Suppose thatGis a group with a normal solvable subgroup H such that G/H is supersolvable. If every subgroup of F(H) of prime order or order 4 is S-quasinormal inG, then G is supersolvable.
ii) If G is solvable and every subgroup of F(G) of prime order or order 4 is S-quasinormal inG, then G is supersolvable.
Corollary 2.1.3(Laue [La]). IfG is solvable and every subgroup ofF(G) of prime order or order 4 is normal in G, then G is supersolvable.
Preliminary results
Lemma 2.1.4. Let F be a saturated formation containing U. Suppose that G is a group with a normal subgroup H such that G/H ∈ F. If every subgroup of H of prime order or order 4 is S-quasinormal inG, then G∈ F.
Proof. See [AsBP, Theorem 1]. In particular, if F =U, a result of Shaalan [Sha, Theorem 3.1] follows.
Lemma 2.1.5 [ACs1, Lemma 2]. If P is an S-quasinormal p-subgroup of G for some prime p, then P is normalized by every p′-element aof G.
Proof. P is subnormal inG(see [We, p. 24, Corollary 6.3]). It suffices to show the assertion for a q-element a of G where q is a prime 6= p. We have a∈ Q for some Q ∈ Sylq(G). Since P is a subnormal and therefore normal Sylow p-subgroup of P Q=QP, the assertion follows.
Lemma 2.1.6 [ACS1, Lemma 3]. Let P be a p-group with p > 2. If a p′ -automorphism a of P centralizes Ω1(P), thena is the identity on P.
Proof. See [Go, p. 184, Theorem 3.20].
Lemma 2.1.7[ACs1, Lemma 4]. LetP be ap-subgroup ofG, wherep >2. Suppose, all subgroups of P of order p are S-quasinormal in G. If a is a p′-element of NG(P)\CG(P), then ainduces in P a fixed point free automorphism.
Proof. Suppose that the action ofaonP is not fixed point free. Then there isc∈P of order p such that ca = c. The subgroup V = hciΩ1(Z(P)) ≤ P is elementary abelian. Moreover, V and all subgroups of V are hai-invariant by Lemma 2.1.5.
Therefore the linear transformation induced by a on V is scalar. It is necessarily the identity asafixesc. By the same argument, for every h∈Ω1(P) witho(h) =p,
we have thata induces the identity on hhiΩ1(Z(P)). Soa centralizes Ω1(P). Now
a∈CG(P) by Lemma 2.1.6.
Proof of Theorem 2.1.1. With the aid of the preceding lemmas we can now prove our theorem.
(a) =⇒(b) If G∈ F, then (b) is true with H = 1.
(b) =⇒ (a) Suppose that the result is false and let G be a counterexample of minimal order. Letpbe the smallest prime dividing|F(H)|and letP ∈Sylp(F(H)).
Clearly P EG. Once we know that G/P inherits the hypothesis (b), we conclude G/P ∈ F by the minimality of |G|. Applying Lemma 2.1.4 we are done.
So we have to prove that (b) is inherited by G/P. Clearly H/P is a solvable normal subgroup of G/P with (G/P)/(H/P) ∼= G/H ∈ F. Let L/P = F(H/P), we have to show that the subgroup of prime order or 4 ofL/P are S-quasinormal inG/P. Let X/P be a subgroup of L/P of prime orderq (of order 2 or 4 ifq = 2).
LetQbe a Sylow q-subgroup ofL. We have:
i) p∤|L/P|, in particular,q6=p:
IfP1/P is the Sylowp-subgroup of the nilpotent characteristic subgroupL/P ofH/P, then P1 EGand thus P1≤F(H), i.e.P1=P.
ii) Qis not cyclic:
Otherwise the Sylowq-subgroupQP/P ofL/P is cyclic. SinceX/P ≤QP/P, thenX/P is a characteristic subgroup ofQP/P, it follows X/P EG/P which certainly implies that X/P is S-quasinormal in G/P.
iii) Lis supersolvable:
This follows by Lemma 2.1.4 sinceL/P is nilpotent, in particular supersolvable, and the subgroups of orderp(of order 2 or 4 ifp= 2) ofP areS-quasinormal inL.
iv) q < p, in particular,p >2 andq ∤|F(H)|:
AsQP/P is the Sylow q-subgroup of L/P, we have that QP EL and QP is supersolvable by iii). If q > p, then Q EQP [We, p. 6, Theorem 1.8], which means that QP = Q×P ≤ F(H). Thus, X ∩Q is S-quasinormal in G by hypothesis and since X = P(X∩Q) it follows that X/P is S-quasinormal inG/P. Soq < p,p >2, andq ∤|F(H)| by choice of p.
v) Qacts faithfully and (elementwise) fixed point freely onP:
LetF(H) =P ×T. So QT ∼=QP/P ×T P/P ≤L/P is nilpotent. Therefore Qcentralizes T. If somea∈Q centralizes alsoP, then a∈CH(F(H))∩Q≤ F(H)∩Q = 1, as H is solvable. So a = 1. Now, by Lemma 2.1.7, every nonidentity element ofQinduces a fixed point free automorphism on P.
vi) Finish of the proof:
By v) and [Go, p. 339, Theorem 3.1], Q is cyclic or generalized quaternion.
Since L is supersolvable, one knows L′ ≤ F(L) = F(H). So L/F(H) is abelian and since q ∤ |F(H)|, it follows that Q ∼= QF(H)/F(H) is abelian, that is cyclic. This is in contradiction with item ii).
A further consequence of Theorem 2.1.1:
Corollary 2.1.8[ACs1, Corollary 4]. Let F be a saturated formation containing U.
Suppose that G is a group with a normal solvable subgroup H such that G/H ∈ F.
ThenG∈ F under either of the following conditions:
a) G is 2-nilpotent and every subgroup of odd prime order of F(H) is S-quasi-normal inG.
b)The Sylow 2-subgroups ofG are abelian and every subgroup of F(H)of prime order isS-quasinormal in G.
Proof. a) Suppose G is 2-nilpotent and let K be the normal 2-complement of G.
Then K1 = K ∩H is the 2-complement of H. Since H/K1 is G-isomorphic with HK/K, we see that H/K1 is in the hypercenter of G/K1. It follows that also G/K1 ∈ F by the hypothesis on F. AlsoF(K1)≤F(H). So we may substituteH byK1 which is of odd order. The result now follows applying Theorem 2.1.1.
b) Let P be an abelian Sylow 2-subgroup of G and put P1 = F(H)∩P. It follows that P1 is in the center of G, since P is abelian and, by hypothesis, every Sylow subgroup of odd order ofG centralizes Ω1(P1) and P1 [Go, p. 178, Theorem 2.4]. The rest of the assertion now follows by the proof of our theorem sinceG/P1
inherits the hypothesis.
Remarks. (a) The theorem is not true for saturated formations, which do not contain U. For example, if F is the saturated formation of all nilpotent groups, then the symmetric group of degree three is a counterexample.
(b) The theorem is not true if we omit the solvability of H. Set G= H×K, whereH= SL(2,5) and K ∈U. Then|F(H)|= 2 andG/H ∼=K ∈U, butG /∈U.
(c) Let G = S3×C, where S3 is the symmetric group of degree three and C a cyclic group of order 3. Clearly, G is supersolvable. It is easy to check that G contains a subgroup of order three fails to be S-quasinormal in G, and so the converse of Corollary 2.1.2 is false.
A corollary of Corollary 2.1.8 is the following result: If G is a solvable group and every subgroup of the Fitting subgroup F(G) of prime order or order 4 is S-quasinormal inG, thenGis supersolvable. Examining the structure of this subclass of supersolvable groups we obtain the following result:
Theorem 2.1.9 [Cs1, Theorem 5]. Let Gbe a solvable group. Then every subgroup ofF(G) of prime order or order 4isS-quasinormal inG if and only ifG=M(N× K), where M is a nilpotent normal subgroup of odd order,N is a nilpotent subgroup, K is a nilpotent Hall subgroup such that M∩(N×K) = 1,F(G)∩N = 1and every minimal subgroup of M is normal in M andS-quasinormal in G.
For the proof we need the following results.
Lemma 2.1.9 (1) [Cs1, Lemma 3]. Let U be a 2-group in a group G, a∈NG(U) with (o(a),2) = 1 and a normalizes every minimal subgroup of U and every cyclic subgroup of order 4. Then a∈CG(U).
Proof. Let h be an element of U of order 4. By the conditions either ha = h or ha = h3. If ha = h3, then a2 ∈ CG(h). Since a is of odd order, we conclude a∈CG(h). Consequently a∈CG(Ω2(U)), which yieldsa∈CG(U).
Lemma 2.1.9 (2)[Cs1, Lemma 4]. LetP be a normalp-subgroup of a solvable group Gwith an odd primep. Suppose every minimal subgroup ofP isS-quasinormal inG.
Then one of the following holds:
(1) every minimal subgroup of P is normal inP.
(2) Q≤CG(P) for every Sylow q-subgroup Q of G withq 6=p.
Proof. Assume there exists an element x0 of P such that o(x0) = p and hx0i is not normal in P. The solvability of Gimplies the existence of a Hall subgroup H with π(H) =π(G)\ {p}. As P is normal in G, hx0i is subnormal inG. From the S-quasinormality of hx0i we easily conclude that H ≤ NG(hx0i). Let H1 be the normal closure of H inG. Obviously H1 ≤NG(hx0i). As hx0i is not normal in G, we find that H1∩P = P0 6= P. Using P ⊳ H1P and H1⊳ H1P we have that the elements of H fix the elements of P/P0 by conjugation. Applying Glauberman’s Theorem [Gla] we get that there exists ν ∈P \P0 such thatH ≤CG(ν). Clearly the elements of H normalize every minimal subgroup of P. Applying Lemma 2.1.7 H ≤CG(P) holds. Let Q be a Sylow q-subgroup ofG with q 6=p. AsQ ≤Hz for somez∈Gand P is normal in G, our statement follows.
Proof of Theorem 2.1.9. Suppose every subgroup of F(G) of order prime or 4 is S-quasinormal inG. It follows from Corollary 2.1.8 thatGis supersolvable, whence G′ is nilpotent and G′ ≤ F(G). Using [Hu2, Satz 3.10 p. 271] G = HF(G) for some nilpotent subgroup H. If P is an arbitrary Sylow subgroup of F(G), denote by P∗ the unique Sylow subgroup of the subgroup HP containing P. Denote by S the set of those Sylow subgroups P of F(G) for which HP is nilpotent. Define S∗ ={P∗|P ∈ S}. SupposeP1∗, P2∗ ∈ S∗. From the above we can easily conclude that P1∗ ≤CG(P2∗). Let K be the direct product of the elements ofS∗. We have
F(G) =M ×(K∩F(G)) and H = N ×(H∩K), for some nilpotent subgroupN and a nilpotent Hall subgroupM. LetB be the Sylow 2-subgroup of F(G) and h an arbitrary element of odd order of H. By the conditions, using Lemma 2.1.9(1), h∈CG(B) holds, whenceBH is nilpotent; consequentlyM is of odd order. We have N K=N×K,M ⊳G,N M ⊳G, and so we findG=M(N×K). AssumeQis a Sylow subgroup for an odd prime of F(G) with H∩Q6= 1. ObviouslyH = (H∩Q∗)×T for some Hall subgroupT of H. Hence T ≤CG(H∩Q) and T ≤NG(Q). By using our hypothesis and applying Lemma 2.1.7 we get T ≤ CG(Q). Thus in this case Q∗ ∈ S∗, that is Q∗ ≤ K. This fact implies M ∩N = 1 and M ∩(N ×K) = 1.
LetRbe an arbitrary Sylow subgroup ofM. By our Lemma 2.1.9(2) every minimal subgroup of R is normal in R and consequently in M too, and is S-quasinormal inG. Obviously F(G)∩N = 1.
Assume conversely G =M(N ×K) has the required properties. Clearly M ≤ F(G)≤M K. Let D be a subgroup of F(G) of prime order or order 4. Supposing D≤M,D isS-quasinormal inG. IfD≤K, using the structure of G, it is easy to
see theS-quasinormality of DinG.
For the description of the structure of Sylow subgroups of M in the previous theorem we can apply the following theorem, so we get a complete characterization:
Theorem 2.1.10 [Cs1, Theorem 4]. Let G be a supersolvable group and P is a normalp-subgroup ofGwithp6= 2. Then every minimal subgroup of P is normal in P and S-quasinormal inGif and only if there is a chain 1 =P0⊳ P1⊳ . . . ⊳ Pk=P with Pi ⊳ G, |Pi/Pi−1|=p for every 1≤i ≤k and Ω1(P) =Pℓ ≤Z(P), for some 1 ≤ ℓ ≤ k. Moreover, for every g ∈ G with (o(g), p) = 1, there exists a natural number tg with 1≤ tg ≤p−1 such that ag =atg, where a is an arbitrary element of D= Xk
i=1(Pi/Pi−1).
For the proof of this latter Theorem 2.1.10 we need our natural factorization of supersolvable groups, so the proof can be founded in Chapter 2.2.
It is meaningful to remove the solvability of H in the hypotheses of Theo-rem 2.1.1, but in this case the Fitting subgroup F(H) will sometimes be a trivial group, while the generalized Fitting subgroupF∗(G) is never trivial if G 6= 1 (see [HuB, X. 13]).
Recall that for any groupG, the generalized Fitting subgroupF∗(G) is the set of all elementsxof Gwhich induce an inner automorphism on every chief factor ofG.
Clearly F∗(G) is a characteristic subgroup ofG. By [Hu2, III. 4.3] F(G)≤F∗(G).
In [LW1, Theorem 3.1] Li and Wang extended our result obtaining the following:
Suppose thatGis a group with a normal subgroupNsuch thatG/N is supersolvable.
If every subgroup of prime order or order 4 of F∗(N) isS-quasinormal in G, then Gis supersolvable. Their proof depends heavily upon our proof.
Later with M. Asaad in [ACs2] we gave a characterization of a groupGunder the assumption that every subgroup ofF∗(G) of prime order or order 4 isS-quasinormal inG.
More precisely, we proved the following:
Theorem 2.1.11 [ACs2, Theorem 1.1]. Let G be a group of composite order such that the quaternion group of order 8 is not involved inG.
Then the following statements are equivalent:
(1) Every subgroup of F∗(G) of prime order is S-quasinormal in G.
(2) G = U W, where U is a normal nilpotent Hall subgroup of odd order and W is a supersolvable Hall subgroup and (|U|,|W|) = 1 and every subgroup of U of prime order isS-quasinormal in G.
(3) G is solvable and every subgroup of F(G) of prime order is S-quasinormal in G.
For the proof of Theorem 2.1.11 we need the following:
Lemma 2.1.12 [ACs2, Lemma 2.1]. Let S ∈ Syl2(G). If every subgroup of S of order 2 is normal in G and the quaternion group of order 8 is not involved in G, then G has a normal 2-complement.
Proof. Let G be a counterexample of minimal order. Then G has not a normal 2-complement, soG contains a minimal non-2-nilpotent subgroupK. By [Hu2, IV.
Satz 5.4] K is a minimal non-nilpotent subgroup of G. By [Hu2, III. Satz 5.2],
|K|= 2nqm for a prime q 6= 2, K has a normal Sylow 2-subgroup K2 and a cyclic Sylowq-subgroupKq, the exponent ofK2 is 2 or 4. Hence if the exponent ofK2 is 2, thenK2 is elementary abelian and soKq EK, a contradiction. Thus the exponent of K2 is 4. By [Hu2, III. Satz 5.2] K2′ = Z(K2) = Φ(K2) ≤ Z(K). Let R be a maximal subgroup of K2′. Then R E K and so K/R is a minimal non-nilpotent group and K2′/R = (K2/R)′ = Z(K2/R) = Φ(K2/R) and |K2′/R| = 2, so K2/R is an extraspecial quaternion free 2-group. Then, by [Go, Chap. 5, Theorem 5.2]
K2/Ris isomorphic to a dihedral group of order 8, soK2/Rcontains a characteristic subgroup of order 4. This implies thatK/Ris nilpotent, a final contradiction.
Lemma 2.1.13 [ACs2, Lemma 2.2]. If every subgroup of prime order of F∗(G) is normal in G and the quaternion group of order 8 is not involved in G, then G is supersolvable.
Proof. If F∗(G) is of odd order, then G is supersolvable by [La]. Thus we may assume that F∗(G) is of even order. By [Hu2, III. Satz 5.3b] F∗(G) is solvable.
Hence using Lemma 2.1.12 we get F∗(G) = F(G). Let S ∈ Syl2(F(G)). Then S E G and SQ is a subgroup of G for every Q ∈ Syl (G) with (|Q|,2) = 1. By Lemma 2.1.12 SQ=S×Q and sinceS EG, it follows easily that every subgroup of order 4 of S isS-quasinormal in G. Therefore every subgroup of prime order or order 4 of F∗(G) is S-quasinormal in G. Let H be any subgroup of order 4 of S.
Then O2(G)≤NG(H)≤G. Hence ifNG(H) =G for allH ≤S,|H|= 4, we have every subgroup of prime order or order 4 of F∗(G) =F(G) is normal in G. Using [La] we get G is supersolvable. Thus there exists H ≤ S such that |H| = 4 and NG(H) < G. ButO2(G)≤ NG(H) < G. Since F∗(O2(G))≤ F∗(G), by [HuB, X.
13, 11 Corollary] we have O2(G) is supersolvable by induction. As G/O2(G) is a 2-group, we haveGis solvable. Now applying Theorem 2.1.1 Gis supersolvable.
Proof of Theorem 2.1.11. (1) =⇒ (3) By induction on the order of G. First we argue thatG is solvable. If every minimal subgroup of F∗(G) is normal in G, then G is supersolvable by Lemma 2.1.13, and so G is solvable. Thus we assume that there exists a subgroup H of prime order p of F∗(G) such that H is not normal in G. By hypothesis H is S-quasinormal in G and so Op(G) ≤ NG(H) < G. But F∗(Op(G)) ≤ F∗(G) by [HuB, X. 13. 11 Corollary] and since every subgroup of F∗(Op(G)) of prime order is S-quasinormal in Op(G), then by induction on |G|, Op(G) is solvable and since G/Op(G) is ap-group, we haveG is solvable.
Now by [HuB, X. 13], F∗(G) =F(G), so every subgroup ofF(G) of prime order isS-quasinormal in G.
(3) =⇒ (2) We argue that every subgroup of F(G) of order 4 is S-quasinormal in G. Let S ∈ Syl2(F(G)). Let M = M/F(G) be a 2′-Hall subgroup of G = G/F(G). Thus |G : M| = 2t. Hence if M = 1, |G| = |G/F(G)| = 2t and so it follows easily that every subgroup ofF(G) of order 4 isS-quasinormal in G.
Thus we may assume that M 6= 1. Consider O2(M) = L. We distinguish the following two cases:
Case 1. L∩S 6= 1. Clearly L E M and every subgroup of L∩S of order 2 is S-quasinormal in L and so easily every subgroup of L∩S of order 2 is normal inL. ObviouslyL∩S ∈Syl2(L), consequently L has a normal 2-complement, i.e.
L= (L∩S)×T. Since TcharLEM thenT EM, usingS EM we getST =S×T. SetQ∈Sylq(G) with q 6= 2, let S0 ≤S be such that |S0|= 4. Clearly Qy ≤T for somey∈G, whenceS0y ≤CG(Qy), and we can conclude S0Qis a subgroup of G.
Case 2. L∩S = 1. Then M = L×S. Let Q ∈Sylq(G) with q 6= 2. Clearly Qz ≤M for somez∈Gand soQz ≤L. Let S0 ≤S be such that |S0|= 4. Clearly S0z ≤S, whence it followsQz ≤CG(S0z), consequentlyQ≤CG(S0).
Thus G is solvable and every subgroup of F(G) of prime order or of order 4 is S-quasinormal in G. Then applying Corollary 2.1.8 we get the supersolvability
ofG. Letpbe the largest prime dividing|G|and P ∈Sylp(G), the supersolvability of G implies P EG. Now let U be a normal nilpotent Hall subgroup of G of odd order such that P ≤ U. Then G = U W, where W is a subgroup of G such that (|U|,|W|) = 1. Clearly U ≤F(G) and hence every subgroup ofU of prime order is S-quasinormal inG.
(2) =⇒(3) By induction on the order ofG. Gis supersolvable by [Sha, Theorem 3.1] and we can concludeG′ ≤F(G). Let M be a maximal subgroup of G contain-ingF(G). ThenM ⊳ GsoF(M) =F(G). Also M =U(M∩W) and so by induction on |G|we have every subgroup of F(M) (=F(G)) of prime order is S-quasinormal inM. Now we argue that every subgroup ofF(G) of prime order is S-quasinormal inG. LetH≤F(G), whereHis of prime order. Since F(G) =U×(F(G)∩W) we have eitherH ≤U orH ≤F(G)∩W. If H≤U then H is S-quasinormal inG by hypothesis. AssumeH ≤F(G)∩W, write |G/M|=r, wherer is a prime number.
Clearly, Hpermutes with every Sylow subgroupQof Gsuch that (|Q|, r) = 1. Now let R ∈ Sylr(G). We claim that HR is a subgroup of G. If |H| = r, then let R∗ ∈Sylr(F(G)). Clearly R∗ EG, consequentlyR∗ ≤R. SinceH ≤R∗ ≤R then HR=RH =R. If |H| 6=r, let |H|=q 6=r and Q∗ ∈ Sylq(F(G)). Then Q∗ EG and Q∗R is a subgroup of G. Hence if R E Q∗R, then Q∗R = Q∗×R, and we can conclude HR is a subgroup of G. Thus we may assume that R is not normal in Q∗R. But Q∗R is supersolvable, so q > r. If Q∗R G, then by the induction on |G|we have that every subgroup of prime order of F(Q∗R) is S-quasinormal in Q∗R and since H ≤Q∗ ≤F(Q∗R) it follows HR is a subgroup of G. IfQ∗R =G, then Q∗ = U, whence H ≤ U, which is a contradiction with H ≤ F(G)∩W and (|U|,|F(G)∩W|) = 1.
(3) =⇒ (1) We have that G is solvable, henceF∗(G) =F(G) by [HuB, X. 13], consequently every subgroup of F∗(G) of prime order is S-quasinormal in G.
Later Y. Li and Y. Wang in [LW2] extended our result Theorem 2.1.1 replac-ing S-quasinormality by requiring that the subgroups of prime order or order 4 of the generalized Fitting subgroup of some normal subgroup areS-quasinormally em-bedded in the whole group. We recall a subgroup H of a group G is said to be S-quasinormally embedded (orπ-quasinormally embedded) inG, if for each primep dividing the order ofHa Sylowp-subgroup ofHis also a Sylow p-subgroup of some S-quasinormal subgroup of G. The proof of Lie and Wang’s theorem is strongly relied on our result.
Recently Asaad and Heliel [AsH] generalized the notion of S-quasinormality, introducing a weaker new embedding property, namely the 3-permutability (or Σ-permutability) of a subgroup of a group. 3 is a complete set of Sylow subgroups of the groupG, if for each prime divisorpof the order ofG 3 contains exactly one Sylow p-subgroup ofG. A subgroup ofGis said to be 3-permutable if it permutes with every
member of 3. Obviously every S-quasinormal subgroup is 3-permutable. Heliel, Xianghua Li and Yangming Li in [HLL] extended our Theorem 2.1.1 replacing the S-quasinormality by 3-permutability. In the proof of their result our Theorem 2.1.1 was used very heavily. Beside our result they needed the classification of finite simple groups. Three years later Li Fang Wang and Yan Ming Wang [WW] gave an
member of 3. Obviously every S-quasinormal subgroup is 3-permutable. Heliel, Xianghua Li and Yangming Li in [HLL] extended our Theorem 2.1.1 replacing the S-quasinormality by 3-permutability. In the proof of their result our Theorem 2.1.1 was used very heavily. Beside our result they needed the classification of finite simple groups. Three years later Li Fang Wang and Yan Ming Wang [WW] gave an