On Nearly S-Permutably Embedded Subgroups of Finite Groups *

Transcription

1 ISSN , Mathematical Notes, 2012, Vol. 91, No. 4, pp Pleiades Publishing, Ltd., Published in Russian in Matematicheskie Zametki, 2012, Vol. 91, No. 4, pp On Nearly S-Permutably Embedded Subgroups of Finite Groups * Kh. A. Al-Sharo ** Al-Bayt University, Jordan Received September 3, 2011 Abstract Let G be a finite group. A subgroup H of G is said to be S-permutable in G if HP = PH for all Sylow subgroups P of G. AsubgroupA of a group G is said to be S-permutably embedded in G if for each Sylow subgroup of A is also a Sylow of some S-permutable subgroup of G. In this paper, we analyze the following generalization of this concept. Let H be a subgroup of a group G. Then we say that H is nearly S-permutably embedded in G if G has a subgroup T and an S-permutably embedded subgroup C H such that HT = G and T H C. We study the structure of G under the assumption that some subgroups of G are nearly S-permutably embedded in G. Some known results are generalized. DOI: /S Keywords: nearly S-permutably embedded subgroup, saturated formation, solvable group, supersolvable group, maximal subgroup. 1. INTRODUCTION Throughout this paper, all groups are finite. An interesting question in finite grouptheory istodetermine the influence of the embedding properties of members of some distinguished families ofsubgroups onthe structure of the group. The present paper adds some results to this line of research. Recall that a subgroup H of a group G is said to be S-permutable, S-quasinormal, or π(g)-quasinormal (see Kegel [1]) in G if HP = PH for all Sylow subgroups P of G. A subgroup A of a group G is said to be S-permutably embedded in G or S-quasinormally embedded in G (see Ballester-Bolinches and Pedraza-Aguilera [2]) if for each Sylow subgroup ofa is also a Sylow of some S-permutable subgroupofg. A subgroupa is said to beweakly S-supplemented ing(see Skiba [3]) if for some subgroup T of G we have AT = G and A T A sg,wherea sg is the subgroup generated by all those subgroups of A which are S-permutable in G. In this paper, we analyze the following generalization of these two concepts. Definition 1.1. Let H be a subgroup of a group G. Then we say that H is nearly S-permutably embedded in G if G has a subgroup T and an S-permutably embedded subgroup C H such that HT = G and T H C. It is clear that every S-permutably embedded subgroup and every weakly S-supplemented subgroup are nearly S-permutably embedded subgroup. The following example shows that, in general, the set of all nearly S-permutably embedded subgroups is wider than the set of all S-permutably embedded subgroups and the set of all weakly S-supplemented subgroups. The text was submitted by the author in English. ** sharo_kh@yahoo.com 470

2 ON NEARLY S-PERMUTABLY EMBEDDED SUBGROUPS OF FINITE GROUPS 471 Example. Let A = C 9 a, wherec 9 is the cyclic group of order 9 and a is an automorphism of C 9 of order 2. Let G = P A, wherep is a simple F 2 [A]-module which is faithful for A. LetH = C a, where C is the subgroup of C 9 of order 3. It is clear that C is S-permutably embedded in G. Moreover, G = HT, where T = PC 9, H T = C. Hence H is a nearly S-permutably embedded subgroup of G. It is clear that a is not S-permutable in G (see Lemma 2.1 (3) and Lemma 2.6 (1) below) and there is no an S-permutable subgroup E such that a isasylowsubgroupofe. HenceH isnotan S-permutablyembedded subgroupofg. Itisalso clear that H sg =1and H has no complement in G. ThusH is not a weakly S-supplemented subgroup of G. In this paper, we prove the following theorems. Theorem A. Let F be a saturated formation containing all supersolvable groups and G a group with normal subgroup E such that G/E F. Suppose that, for every noncyclic Sylow subgroup P of E, every cyclic subgroup of P of prime order or order 4 is nearly S-permutably embedded in G. ThenG F. Theorem B. Let F be a saturated formation containing all supersolvable groups and G a group with normal subgroup E such that G/E F. Suppose that, for every noncyclic Sylow subgroup P of E, every maximal subgroup of P is nearly S-permutably embedded in G. Then G F. The main stage in the proof of Theorem B is the following result, which is of independent interest. Theorem C. Let G be a group, p the smallest prime dividing G and P asylowp-subgroup of G. If every maximal subgroup of P is nearly S-permutably embedded in G, theng is p-nilpotent. We prove Theorems A, B, and C in Sec. 3. But before, in Sec. 2, we describe the most general properties of the nearly S-permutably embedded subgroups. In Sec. 4 we consider some applications of these theorems. All unexplained notation and terminology are standard. The reader is referred to [4] and [5]. 2. PRELIMINARIES In our proofs, we shall need the following known properties of S-permutable, S-permutably embedded and subnormal subgroups. Lemma 2.1 ([1]). Let G be a group and H K G. (1) If H is S-permutable in G, thenh is S-permutable in K. (2) Suppose that H is normal in G. Then K/H is S-permutable in G/H if and only if K is S-permutable in G. (3) If H is S-permutable in G, thenh is subnormal in G. From Lemma 2.1 (3), we obtain the following statement. Lemma 2.2. If H is an S-permutable subgroup of a group G and H is a p-group for some prime p, then O p (G) N G (H). Lemma 2.3. Let G be a group and A, B G. (1) If A is S-permutable in G, thena B is S-permutable in B [6]. (2) If A and B are S-permutable in G, thena B is S-permutable in G [1]. (3) If A is S-permutable in G, thena/a G is nilpotent [6].

3 472 AL-SHARO Lemma 2.4 ([2]). Let G be a group and H K G. (1) If H is S-permutably embedded in G, thenh is S-permutably embedded in K. (2) If H is normal in G and E is an S-permutably embedded subgroup of G, theneh is S-permutably embedded in G and EH/H is S-permutably embedded in G/H. From Lemma 2.3 (2), we obtain the following statement. Lemma 2.5. Let H be an S-permutably embedded subgroup of a group G. IfH O p (G) for some prime p,thenh is S-permutable in G. Lemma 2.6 ([7]). Let G be a group and A G. (1) If A is subnormal in G and A is a π-subgroup of G, thena O π (G). (2) If A is subnormal in G and A is nilpotent, then A F (G). Lemma 2.7. Let G be a group, H K G, andn a normal subgroup of G. (1) If H is nearly S-permutably embedded in G, thenh is nearly S-permutably embedded in K. (2) If H is normal in G and K is nearly S-permutably embedded in G, thenk/h is nearly S-permutably embedded in G/H. (3) Suppose that ( N, H ) =1andH is S-permutably embedded in G. ThenHN/N is nearly S-permutably embedded in G/N. Proof. (1) Let T and C H besubgroupsofg such that HT = G, C iss-permutably embedded in G and H T C. Then K = H(T K) and (T K) H = T H C, where C is S-permutably embedded in K by Lemma 2.4 (1). Hence we have (1). (2) Let T be a subgroup of G such that KT = G and K T C, where C is S-permutably embedded in G. Then (K/H)(HT/H)=G/H and K/H HT/H = H(K T )/H CH/H NH/N, where CH/H is S-permutably embedded in G/H by Lemma 2.4 (2). Hence we have (2). (3) Let T be a subgroup of G such that HT = G and H T C, where C is S-permutably embedded in G. Since( N, H ) =1,wehaveN T.HenceTN/N = T/N, so T/N HN/N = N(T H)/N NC/N NH/N, and NC/N is S-permutably embedded in G/N by Lemma 2.4 (2). Hence we have (3). Lemma 2.8. Let P 1 be an elementary Abelian normal subgroup of a group G. Assume that every maximal subgroup of P is nearly S-permutably embedded G. Then some maximal subgroup of P is normal in G. Proof. Suppose that this lemma is false and consider a counterexample (G, P ) for which G P is minimal. Let N be a minimal normal subgroup of G contained in P. The hypothesis holds for (G/N, P/N), so some maximal subgroup M/N of P/N is normal in G/N by the choice of (G, P ). Hence the maximal subgroup M of P is normal in G, which contradicts the choice of (G, P ). HenceP = N is a minimal normal subgroup of G. Suppose that some maximal subgroup M of P is not S-permutably embedded in G. Then, for some proper subgroupt of G, wehave MT = G and M T N. Hence 1 P T P.ButP Tisnormal in G, which contradicts the minimality of N. Therefore, every maximal subgroup of P is S-permutably embedded in G. Hence every maximal subgroup of P is S-permutable in G by Lemma 2.5 and so some maximal subgroup of P is normal in G by Lemma 2.11 in [3].

4 ON NEARLY S-PERMUTABLY EMBEDDED SUBGROUPS OF FINITE GROUPS 473 Lemma 2.9 ([8, Chap. VI, Theorem 24.2]). Let F be a saturated formation and G agroupsuchthat G F is solvable and every maximal subgroup of G not containing G F belongs to F. (a) P = G F is a p-group for some prime p and P is of exponent p or exponent 4(if P is a non-abelian 2-group). (b) P/Φ(P ) is a chief factor of G and (P/Φ(P )) (G/C G (P/Φ(P ))) / F. (c) If P is Abelian, then Φ(P )=1. Lemma 2.10 ([3, Lemma 2.16]). Let F be a saturated formation containing U and G be a group with a normal subgroup E such that G/E F.IfE is cyclic, then G F. Lemma 2.11 ([9, Chap. V, Theorem 9.1]). If G is a supersolvable group, then G is p-nilpotent where p is the smallest prime dividing G. Lemma 2.12 ([8, Chap. II, 7.9]). Let P be a nilpotent normal subgroup of a group G. IfP Φ(G) = 1, thenp is the direct product of some minimal normal subgroups of G. The following lemma is well known. Lemma Let A, B G and G = AB. Then (1) there are Sylow p-subgroups A p, B p, andg p of A, B, andg, respectively, such that G p = A p B p ; (2) G = AB x for all x G. 3. PROOFS OF THEOREMS A, B, AND C 3.1. Proof of Theorem A Suppose that this theorem is false and consider a counterexample (G, E) forwhich G E is minimal. Let p be the smallest prime dividing E and P asylowp-subgroup of E. We divide the proof in several steps, treating them as separate assertions. Step 1. If X is a Hall subgroup of E, the hypothesis is still true for (X, X). If, in addition, X is normal in G, then the hypothesis also holds for (G/X, E/X). Proof. Indeed in view of Lemma 2.7 (1), the hypothesis holds for (X, X). On the other hand, if X is normal in G and Q/X is a noncyclic Sylow q-subgroup of G/X, then for some Sylow q-subgroup Q 1 of G we have Q/X = Q 1 X/X Q 1. Hence, in view of the isomorphism (G/X)/(E/X) G/E F, the hypothesis also holds for (G/X, E/X) (by Lemma 2.7 (3)). Step 2. If X is a nonidentity normal Hall subgroup of E, thenx = E. Proof. Since X is a characteristic subgroup of E, it is normal in G and by (1) the hypothesis is still true for (G/X, E/X). HenceG/X F by the choice of G. Thus, the hypothesis is still true for (G, X) and so X = E (by the choice of (G, E)). Step 3. E = G F. Proof. Since G/E F,wehaveG F E. On the other hand, the hypothesis holds for (G, G F ) by Lemma 2.7 (1). Hence E = G F by the choice of (G, E). Step 4. P E.

5 474 AL-SHARO Proof. Suppose that P = E. Since the formation F is saturated and G/E F, it follows that P Φ(G). LetM be any maximal subgroup of G not containing E. Then M/E M EM/E = G/E F. Hence the hypothesis holds for (M,E M) by Lemma 2.7 (1). Hence M F by the choice of (G, E). Every maximal subgroup of G not containing E = G F belongs to F. Therefore, by Lemma 2.8, P is a group of exponent p or exponent 4 (ifp is a non-abelian2-group). Moreover, P/Φ(P ) is a chief factor of G and (P/Φ(P )) (G/C G (P/Φ(P ))) / F. Let Φ=Φ(P ), X/Φ be a minimal subgroup of P/Φ, x X \ Φ, andl = x. Then L = p or L =4. Hence L is nearly S-permutably embedded in G. Suppose that L is not S-permutably embedded in G. Then, for some proper subgroup T of G, wehavelt = G. SinceΦ Φ(G), wehaveφt <G.Since the maximal subgroup of L is contained in Φ,wehave G :ΦT = p. Hence P/Φ = G :ΦT = p, so G/Φ F by Lemma Thus, by the result of Step 3, we have P = G F Φ <P. This contradiction shows that L is S-permutably embedded in G. Therefore L is S-permutable in G by Lemma 2.5. Hence LΦ/Φ =X/Φ is S-permutable in G/Φ by Lemma 2.1 (2). Therefore, every minimal subgroup of P/Φ is S-permutable in G/Φ, andso P/Φ = p by Lemma 2.12 in [3]; a contradiction. Hence P E. Step 5. E = G. Proof. Suppose that E<G.Thehypothesisistruefor(E,E) by Step 1, so E is supersolvable by the choice of (G, E). HenceE is p-nilpotent by Lemma Let V be a Hall p -subgroup of E. ThenV is normal in E, sov =1by 2 and hence E = P, which contradicts 4. Hence we prove the desired statement. The final contradiction. In view of 2, G is not p-nilpotent. Therefore G has a p-closed Schmidt subgroup H = H p H q [9, Chap. IV, Theorem 5.4]. Without loss we may assume that H p P. By Lemma 2.9, H p /Φ(H p ) is a noncentral chief factor of H and H p is a group of exponent p or exponent 4 (if p =2and H p is non-abelian). Hence H p /Φ(H p ) >p,sincep is the smallest prime dividing H. On the other hand, by Lemma 2.7 (1) every cyclic subgroup of H of prime order or order 4 is nearly S-permutably embedded in H. Hence, as in the proof of Step 4, one can show that H p /Φ(H p ) = p. This contradiction completes the proof of the result Proof of Theorem C Suppose that this theorem is false and let G be a counterexample of minimal order. Just as in the preceding theorem, we divide the proof into several separate steps. Step 1. O p (G) =1. Proof. Let D = O p (G). The hypothesis is still true for G/D by Lemma 2.7 (3), so in the case where D 1, G/D is p-nilpotent by the choice of G. HenceG is p-nilpotent; a contradiction. Thus, we have O p (G) =1. Step 2. If P V < G,thenV is p-nilpotent. Proof. Indeed, by Lemma 2.7 (1) the hypothesis holds for V,soV is p-nilpotent by the choice of G.

6 ON NEARLY S-PERMUTABLY EMBEDDED SUBGROUPS OF FINITE GROUPS 475 Step 3. O p (L) =1for any S-permutable subgroup L of G. Proof. Indeed, by Lemma 2.1 (3), L is subnormal in G, soo p (L) is subnormal in G. O p (L) O p (G) =1by Lemma 2.6 (1). But then Step 4. If N is an Abelian minimal normal subgroup of G, theng/n is p-nilpotent. Proof. In view of Step 1, N is a p-group, so N P. Thus,thehypothesisistrueforG/N by Lemma 2.7 (2). Hence we have the desired statement by the choice of G. Step 5. G is p-solvable. Proof. In view of Step 4, we need only to show that G has an Abelian minimal normal subgroup. Suppose that this is false. Then p =2by Odd Theorem, and by Lemmas 2.1 (3) and 2.6 (1), every nonidentity subgroup of P is not S-permutable in G. Hence for any maximal subgroup M of P such that 1 C M for some S-permutably embedded subgroup C and, for any S-permutable subgroup W of G such that C = W 2 Syl 2 (W ), we have C W.Moreover,W G 1. Indeed, if W G =1,thenW is nilpotent by Lemma 2.3 (3). Hence O 2 (W ) 1, which contradicts Step 3. Note also that, for any minimal normal subgroup N of G, we have NP = G; otherwise, N is 2-nilpotent by Step 2. But this contradicts Step 3. Hence N is the only minimal normal subgroup of G and so N P = N C. Let M 1 be a maximal subgroup of P and T 1 a subgroup of G such that M 1 T 1 = G and M 1 T 1 C 1 for some S-permutably embedded in G subgroup C 1 M 1. Suppose that C 1 =1. Then T 1 is a complement of M 1 in G, sot 1 is 2-nilpotent by [10, Chap. 7, Theorem 6.1]. Without loss of generality, we may assume T 1 = N G (H 1 ) for some Hall 2 -subgroup H 1 of G. It is clear that H 1 N. By [11], every two Hall 2 -subgroups of G are conjugate and so, by Frattini s argument, G = NT 1.Hence P =(P N)(P T 1 ) by Lemma 2.13 (1). It is clear that P T 1 P. Hence we can choose a maximal subgroup M 2 in P containing P T 1. By assumption, G = M 2 T 2,whereM 2 T 2 C 2 for some S-permutably embedded in G subgroup C 2 M 2. First suppose that C 2 =1,soT 2 is 2-nilpotent. Again, we can assume that T 2 = N G (H 2 ) for some Hall 2 -subgroup H 2 of G. By [11], we have H x 1 = H 2 for some x G. Therefore, by Lemma 2.13 (2), and G = M 1 T 1 = M 2 T 2 = M 2 T 1 x = M 2 T 1 P = M 2 (P T 1 )=M 2 ; a contradiction. Therefore C 2 1.LetW be an S-permutable subgroup of G such that C 2 Syl p (W ). Then N W and C 2 N = P N, which implies P =(P N)(P T 1 )=(C 2 N)(P T 1 ) M 2 ; a contradiction. Therefore, every maximal subgroup M 1 of P has a nonidentity S-permutably embedded in G subgroup C 1. But then, from the above, we have N P = N M 1 M 1.HenceN P Φ(P ), and N is 2-nilpotent by Theorem 4.7 in [9]; a contradiction. Hence we have the desired condition.

7 476 AL-SHARO The final contradiction. Let N be any minimal normal subgroup of G. Then, in view of Steps 3 and 5, N is a p-group, and hence G/N is p-nilpotent by Step 4. Therefore N is a unique minimal normal subgroup of G and N Φ(G). HenceG is a primitive group, so N = C G (N) =F (G) by [4, Chap. A, Theorem 17.2]. Let P 1 be a maximal subgroup of P such that P 1 N = P, M a maximal subgroup of G such that G = N M. SinceP 1 is nearly S-permutably embedded in G, so there is a subgroup T of G such that P 1 T = G and P 1 T C for some S-permutably embedded in G subgroup C P 1. Suppose that C =1.ThenT is a complement of P 1 in G, so T p = p, wheret p Syl p (T ). HenceT is p-nilpotent, so T p T,whereT p isahallp - subgroup of T. Without loss of generality, we may assume that T p M, so G/N M N G (T p ). Hence G = T p,t p N G (T p ), which contradicts Step 1. Hence C 1.LetW be an S-permutable subgroup of G such that C Syl p (W ). Suppose that C = W.Then N C G = C PT p = C P P 1 ; a contradiction. Hence C W, and so, in view of Step 1 and Lemma 2.1 (3), we have W G 1.Hence N C P 1 and, therefore, P 1 = P 1 N = P ; a contradiction. The theorem is proved Proof of Theorem B Suppose that this theorem is false and consider a counterexample (G, E) forwhich G E is minimal. Let p be the smallest prime dividing E, andletp beasylowp-subgroup of E. Step 1. If X is a nonidentity normal Hall subgroup of E, thenx = E. Proof. See Step 1 in the proof of Theorem A. Step 2. P E. Proof. Suppose that P = E. Let N be any minimal normal subgroup of G contained in P. Then the hypothesis holds for (G/N, P/N). Hence G/N F by the choice of (G, E) =(G, P ). Hence N is the only minimal normal subgroup of G contained in P, N Φ(G) and N >pby Lemma Hence Φ(G) P =1, so, in view of Lemma 2.12, we have N = P.Butthen N = p by Lemma 2.8; a contradiction. Hence we have the desired statement. Step 3. E is not p-nilpotent. Proof. Indeed, if E is p-nilpotent and V is a Hall p -subgroup of E, thenv =1by Step 1, which contradicts Step 2. Step 4. E = G. Proof. Suppose that E<G. The hypothesis holds for (E,E), soe is supersolvable by the choice of (G, E). HenceE is p-nilpotent by Theorem C, which contradicts Step 3. Hence E = G. The final contradiction. In view of Step 4 and Theorem C, G is p-nilpotent which contradicts Step 3. The theorem is proved.

8 ON NEARLY S-PERMUTABLY EMBEDDED SUBGROUPS OF FINITE GROUPS APPLICATIONS OF THEOREMS A, B, AND C From Theorem A, we obtain the following statement. Corollary 4.1 (Buckley [12]). Let G be a group of odd order. If all subgroups of G of prime order are normal in G, theng is supersolvable. Corollary 4.2 (Shaalan [13]). Let G be a group and E a normal subgroup of G with supersolvable quotient. Suppose that all minimal subgroups of E and all its cyclic subgroups with order 4 are S-permutable in G. ThenG is supersolvable. Corollary 4.3 (Ballester-Bolinches, Pedraza-Aguilera [14]). Let F be a saturated formation containing all supersolvable groups and G a group with a solvable normal subgroup E such that G/E F. If all minimal subgroups and all cyclic subgroups with order 4 of E are S-permutable in G,thenG F. A subgroup H of a group G is said to c-normal in G [15] if G has a normal subgroup T such that HT = G and H T H G. Corollary 4.4 (Wang [15]). If all subgroups of G of prime order or order 4 are c-normal in G,thenG is supersolvable. Corollary 4.5 (Ballester-Bolinches, Wang [16]). Let F be a saturated formation containing all supersolvable groups. If all minimal subgroups and all cyclic subgroups with order 4 of G F are c-normal in G, theng F. Corollary 4.6 (Ramadan, Azzat Mohamed, and Heliel [17]). Let F be a saturated formation containing all supersolvable groups, G be a group with normal subgroup E such that G/E F.If all minimal subgroups and all cyclic subgroups with order 4 of E are c-normal in G,thenG F. A subgroup A is said to be c-supplemented in G [18] if for some subgroup T of G we have AT = G and A T A G. Corollary 4.7 (Ballester-Bolinches, Wang, and Guo [18], Wang and Li [19]). Let F be a saturated formation containing all supersolvable groups and G a group with a normal subgroup E such that G/E F. If all minimal subgroups and all cyclic subgroups with order 4 of E are c-supplemented in G,thenG F. Corollary 4.8 (Li and Wang [20]). If all subgroups of G of prime order or order 4 are S-permutably in G,thenG is supersolvable. From Theorem B, we obtain the following statement. Corollary 4.9 (Srinivasan [21]). If the maximal subgroups of the Sylow subgroups of G are normal in G,thenG is supersolvable. Corollary 4.10 (Wang [15]). If the maximal subgroups of the Sylow subgroups of G are c-normal in G,thenG is supersolvable. Corollary 4.11 (Asaad [22]). Let F be a saturated formation containing all supersolvable groups and G a group with a normal subgroup E such that G/E F. If every maximal subgroup of any Sylow subgroup of E is S-permutable in G,thenG/E F. Corollary 4.12 (Asaad and Heliel [23]). Let F be a saturated formation containing all supersolvable groups and G a group with a normal subgroup E such that G/E F. If every maximal subgroup ofany Sylow subgroupof E is S-permutably embedded, in G, theng/e F. Corollary 4.13 (Ballester-Bolinches and Guo [24]). Let F be a saturated formation containing all supersolvable groups and G a group with a normal subgroup E such that G/E F. If the maximal subgroups of the Sylow subgroups of E are c-supplemented in G, theng F. Corollary 4.14 (Ramadan, Azzat Mohamed, and Heliel [17]). Let F be a saturated formation containing all supersolvable groups, G be a group with normal subgroup E such that G/E F. If the maximal subgroups of the Sylow subgroups of E are c-normal in G,thenG F.

9 478 AL-SHARO REFERENCES 1. O. H. Kegel, Sylow-Gruppen und Subnormalteiler endlicher Gruppen, Math. Z. 78 (1), (1962). 2. A. Ballester-Bolinches and M. C. Pedraza-Aguilera, Sufficient conditions for supersolvability of finite groups, J. Pure Appl. Algebra 127 (2), (1998). 3. A. N. Skiba, On weakly s-permutable subgroups of finite groups, J. Algebra 315 (1), (2007). 4. K. Doerk and T. Hawkes, Finite Soluble Groups, inde Gruyter Exp. Math. (Walter de Gruyter, Berlin, 1992), Vol A. Ballester-Bolinches and L. M. Ezquerro, Classes of Finite Groups, in Math. Appl. (Springer) (Springer-Verlag, Dordrecht, 2006), Vol W. E. Deskins, On quasinormal subgroups of finite groups, Math. Z. 82 (2), (1963). 7. H. Wielandt, Subnormal subgroups and permutation groups, in Lectures given at the Ohio State University (Columbus, OH, 1971). 8. L. A. Shemetkov, Formation of Finite Groups, incontemporary Algebra (Nauka, Moscow, 1978) [in Russian]. 9. B. Huppert, Endliche Gruppen. I, indie Grundlehren Math. Wiss. (Springer-Verlag, Berlin, 1967), Vol D. Gorenstein, Finite Groups (Harper & Row Publ., New York, 1968). 11. F. Gross, Conjugacy of odd order Hall subgroups, Bull. London Math. Soc. 19 (4), (1987). 12. J. Buckley, Finite groups whose minimal subgroups are normal, Math. Z. 116 (1), (1970). 13. A. Shaalan, The influence of π-quasinormality of some subgroups on the structure of a finite group, Acta Math. Hungar. 56 (3-4), (1990). 14. A. Ballester-Bolinches and M. C. Pedraza-Aguilera, On minimal subgroups of finite groups, Acta Math. Hungar. 73 (4), (1996). 15. Y. Wang, c-normality of groups and its properties, J. Algebra 180 (3), (1996). 16. A. Ballester-Bolinches and Y. Wang, Finite groups with some C-normal minimal subgroups, J. Pure Appl. Algebra 153 (2), (2000). 17. M. Ramadan, M. Ezzat Mohamed, and A. A. Heliel, On c-normality of certain subgroups of prime power order of finite groups, Arch. Math. (Basel) 85 (3), (2005). 18. A. Ballester-Bolinches, Y. Wang, and X. Guo, c-supplemented subgroups of finite groups, Glasgow Math. J. 42 (3), (2000). 19. Y. Wang, Y. Li, and J. Wang, Finite groups with c-supplemented minimal subgroups, Algebra Colloq. 10 (3), (2003). 20. Y. Li and Y. Wang, On π-quasinormally embeddedsubgroupsoffinitegroups, J. Algebra 281 (1), (2004). 21. S. Srinivasan, Two sufficientconditions for supersolvabilityof finite groups, Israel J. Math. 35 (3), (1980). 22. M. Asaad, On maximal subgroups of Sylow subgroups of finite groups, Comm. Algebra 26 (11), (1998). 23. M. Asaad and A. A. Heliel, On S-quasinormally embedded subgroups of finite groups, J. Pure Appl. Algebra 165 (2), (2001). 24. A. Ballester-Bolinches and X. Guo, On complemented subgroups of finite groups, Arch. Math. (Basel) 72 (3), (1999).