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Some notes on the minimal subgroups of Fitting subgroups of finite groups
Journal of Pure and Applied Algebra 171 (2002) 289–294
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Some notes on the minimal subgroups of Fitting subgroups of 'nite groups a Department
b Department
Li Yangminga; b; ∗
of Mathematics, Zhongshan University, Guangzhou 510275, China of Mathematics, Guangdong College of Education, Guangzhou 510303, China
Received 5 January 2001; received in revised form 4 May 2001 Communicated by G.M. Kelly
Abstract Let G be a 'nite group. The question of how the properties of its minimal subgroups in4uence the structure of G is of considerable interest for some scholars. In this paper, we use the c-normal condition on minimal subgroups of Fitting subgroups to characterize the structure of G through c 2002 Elsevier Science B.V. All the theory of formations, generalizing some known results. rights reserved. MSC: 20D10; 20D30
1. Introduction and preliminaries All groups considered in this note will be 'nite. We use conventional notions and notations, as in [8]. Fix a group G. A minimal subgroup of G is a subgroup of prime order. How minimal subgroups can be embedded in G is a question of particular interest in studying the structure of G (see [1– 6,10 –14]). Ballester-Bolinches and Wang in [5] study the in4uence of c-normality of the minimal subgroups on the structure of G through the theory of formations and obtain many new results which make some classical and recent results as particular cases. In this note, we go further by studying the in4uence of the minimal subgroups of Fitting subgroups of G, improving some results of [5,12]. ∗
Corresponding author. Department of Mathematics, Guangdong College of Education, Guangzhou 510303, China. E-mail address: [email protected] (Li Yangming).
c 2002 Elsevier Science B.V. All rights reserved. 0022-4049/02/$ - see front matter PII: S 0 0 2 2 - 4 0 4 9 ( 0 1 ) 0 0 1 2 6 - 8
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Li Yangming / Journal of Pure and Applied Algebra 171 (2002) 289–294
Let F be a class of groups, we call F a formation provided that (i) if G ∈ F and H / G, then G=H ∈ F, and (ii) if G=M and G=N are in F, then G=(M ∩ N ) is in F for normal subgroups M; N of G. A formation F is said to be saturated if G=(G) ∈ F implies that G ∈ F (see [8, Chapter VI]). Throughout this paper U; N will denote the class of all supersolvable groups and the class of all nilpotent groups, respectively. Clearly, U and N are formations. Since a group G is supersolvable iH G=(G) is supersolvable (see [8, p. 713, Satz 8.6]), it follows that U is saturated. N is saturated too (see [8, p. 220, Satz 3.5]). Let F be a saturated formation. A normal subgroup N of a group G is said to be F-hypercentral in G provided N has a chain of subgroups 1 = N0 / N1 / · · · / Nr = N such that Ni+1 =Ni is an F-central chief factor of G. The product of all F-hypercentral subgroups of G is again an F-hypercentral subgroup of G. It is denoted by ZF (G) and called the F-hypercenter of G. For a formation F, each group has a smallest normal subgroup N such that G=N is in F. This uniquely determined normal subgroup of G is called the F-residual subgroup of G and is denoted by G F . For the formation N, some authors prefer to write K∞ (G) as G N and Z∞ (G) as ZN (G). (see [8, Satz III 2.5(d)]). We say, following Wang [11] that a subgroup H of a group G is c-normal in G if there exists a normal subgroup N of G such that G = HN and H ∩ N 6 HG = CorG (H ). Let x be an element of G, we say that x is c-normal in G if x is c-normal in G. First, we give some known results. Lemma 1. Let G be a group. Then (1) If H is normal in G; then H is c-normal in G. (2) If H is c-normal in G; H 6 K 6 G; then H is c-normal in K. (3) Let K / G and K 6 H . Then H is c-normal in G if and only if H=K is c-normal in G=K. (4) Let be a set of primes; H is a normal -subgroup of G and T a -subgroup of G. If T is c-normal in G; then TH=H is c-normal in G=H . (5) If P be a minimal normal p-subgroup of G and x ∈ P is c-normal in G; then P = x. Proof. Conditions (1) – (3) can be found in [11, Lemma 2:1], (4) in [9, Lemma 2:4], and (5) in [12, Lemma 2:2(1)]. Lemma 2. Let P be a normal p-subgroup of G and H / G. Then F(HP) = F(H )P. Proof. See [9, p. 522, Lemma 3:1]. Lemma 3. Let F be a saturated formation containing N. Suppose that G is a group with a normal subgroup H such that G=H ∈ F. Then G ∈ F if and only if every
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element of order 4 of H is c-normal in G and x lies in the F-hypercenter ZF (G) of G for every element x of prime order of H . Proof. See [5, Theorem 3:1]. Lemma 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 c-normal in G; then G ∈ F. Proof. See [5, Theorem 3:4]. 2. Results Theorem 1. Let F be a saturated formation containing N 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 every element of order 4 of F(H ) is c-normal in G and x lies in the F-hypercenter ZF (G) of G for every element x of prime order of F(H ). Proof. (a) ⇒ (b) If G ∈ F, then (b) is true with H = 1. (b) ⇒ (a) Suppose that the results are false and let G be a counterexample of minimal order. Obviously, H = 1, then F(H ) = 1 by the hypothesis of the solvability of H . Let p be a prime dividing |F(H )| and let P ∈ Sylp (F(H )). Clearly, P / G. Once we know that G=P inherits the Hypothesis (b), we conclude G=P ∈ F by the minimality of |G|. Applying, Lemma 3, 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. Since F(H=P) = F(H )=P by Lemma 2, we have to show that the cyclic subgroup of order 4 of F(H )=P are c-normal in G=P and every subgroup of prime order of F(H )=P lies in the F-hypercenter ZF (G=P) of G=P. If p = 2, then there exists no element of order 4 in F(H )=P. Hence suppose p = 2. Let X=P be a cyclic subgroup of F(H )=P of order 4 . By the Schur– Zassenhaus Theorem [7, p. 221, Theorem 2:1], X = xP where x is an element of order 4 of F(H ). Since x is c-normal in G by hypothesis, X=P is c-normal in G=P by Lemma 1(4). Since ZF (G)P=P 6 ZF (G=P), it is easy to see that every subgroup of prime order of F(H )=P lies in the F-hypercenter ZF (G=P) by the hypothesis on the element of prime order of F(H ). These 'nish the proof. Remark 1. An equivalent form of Theorem 1 is as follows (comparing with [5, Theorem 3:1]): Theorem 1 . Let F be a saturated formation containing N and let G be a group. Equivalent are (a) G ∈ F.
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(b) G F is solvable and every element of order 4 of F(G F ) is c-normal in G and x lies in the F-hypercenter ZF (G) of G for every element x of prime order of F(G F ). Remark 2. If F = N, then Theorem 1 coincides with [12, Theorem 3:1]. Theorem 2. Let F be a saturated formation containing N and let G be a group with a normal solvable subgroup H in G such that G=H ∈ F. Then G ∈ F under either of the following: (a) G is 2-nilpotent and x lies in the F-hypercenter ZF (G) of G for every element x of prime order of F(H ). (b) The Sylow 2-subgroups of H are abelian and x lies in the F-hypercenter ZF (G) of G for every element x of prime order of F(H ). Proof. (a) Suppose G is 2-nilpotent, then H is also 2-nilpotent. Denote K is the 2-complement of H , then K / G. Since (G=K)=(H=K) ∼ = G=K ∈ F, and F(H=K) = H=K is a 2-subgroup, it follows that G=K ∈ F by the hypotheses on F and induction on |G|. Also, F(K) 6 F(H ). So we may substitute H by K which is of odd order. Then, the result now follows applying Theorem 1. (b) Let P be an abelian Sylow 2-subgroup of H and put P1 = F(H ) ∩ P. For any x in 1 (P1 ), consider the minimal 2-subgroup P2 of G contained in xG = xg | g ∈ G which is the minimal normal subgroup of G containing x. Then P2 6 F(H ). By the hypothesis of F(H ) and Lemma 1(5), we have that P2 = xg for some g ∈ G. Hence xg lies in the center of G, so does x. It follows that 1 (P1 ) ⊆ Z(G). By [7, p. 178, Theorem 2:4], every element of odd order centralizes P1 , thus P1 is in the center of G since P is abelian. Since G=P1 inherits the hypothesis, G=P1 ∈ F. Thus G ∈ F by the proof of our Theorem 1. Remark 3. Theorem 2 is analogous to [4, Theorems 3 and 4]. Theorem 3. Let F be a saturated formation containing 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 order 4 of the Fitting subgroup F(H ) are c-normal in G. Proof. (a) ⇒ (b) If G ∈ F, then (b) is true with H = 1. (b) ⇒ (a) Suppose that the results are false and let G be a counterexample of minimal order. Obviously, H = 1, then F(H ) = 1 by the hypothesis of the solvability of H . Let p be a prime dividing |F(H )| and let P ∈ Sylp (F(H )). Clearly, P / G. Once we know that G=P inherits the Hypothesis (b), we conclude G=P ∈ F by the minimality of |G|. Applying, Lemma 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. Since F(H=P) = F(H )=P by
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Lemma 2, we have to show that the subgroup of prime order or 4 of F(H )=P are c-normal in G=P. Let X=P be a subgroup of F(H )=P of prime order q (of order 2 or 4 if q = 2). Obviously, q = p. By the Schur–Zassenhaus Theorem [7, p. 221, Theorem 2:1], X = xP where x is an element of prime order or 4 of F(H ). Since x is c-normal in G by hypothesis, thus X=P is c-normal in G=P by Lemma 4(4). These 'nish the proof. Remark 4. An equivalent form of Theorem 3 is as follows (comparing with [5, Theorem 3:4]): Theorem 3 . Let F be a saturated formation containing U and let G be a group. Equivalent are (a) G ∈ F. (b) G F is solvable and every element of prime order or order 4 of F(G F ) is c-normal in G. Remark 5. Theorem 3 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. Remark 6. Theorem 3 is not true if we omit the solvability of H . Set G = H × K, where H = SL(2; 5) and K ∈ U. Then |F(H )| = 2 and G=H ∼ = K ∈ U, but G ∈ U. Remark 7. If F = U, then Theorem 3 coincides with [12, Theorem 3:2]. Theorem 4. Let F be a saturated formation containing U and let G be a group with a normal solvable subgroup H in G such that G=H ∈ F. Then G ∈ F under either of the following: (a) G is 2-nilpotent and every subgroup of odd prime order of F(H ) is c-normal in G. (b) The Sylow 2-subgroups of G are abelian and every subgroup of F(H ) of prime order is c-normal in G. Proof. (a) Suppose G is 2-nilpotent, then H is also 2-nilpotent. Denote K is the 2-complement of H , then K / G. Since (G=K)=(H=K) ∼ = G=K ∈ F, and F(H=K) = H=K is a 2-subgroup, it follows that G=K ∈ F by the hypotheses on F and induction on |G|. Also, F(K) 6 F(H ). So we may substitute H by K which is of odd order. Then, the result now follows applying Theorem 3. (b) Let P be an abelian Sylow 2-subgroup of G and put P1 = F(H ) ∩ P. For any x in 1 (P1 ), consider the minimal 2-subgroup P2 of G contained in xG = xg | g ∈ G which is the minimal normal subgroup of G containing x. Then P2 6 F(H ). By the hypothesis of F(H ) and Lemma 1(5), we have that P2 = xg for some g ∈ G. Hence xg lies in the center of G, so does x. It follows that 1 (P1 ) ⊆ Z(G). By [7, p. 178, Theorem 2:4], every element of odd order centralizes P1 , thus P1 is in the center of
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G since P is abelian. Since G=P1 inherits the hypothesis, G=P1 ∈ F. Thus G ∈ F by the proof of our Theorem 3. Acknowledgements The author is very much indebted to Professor Wang Yanming for his encouragement and advice. The research is supported by the Foundation of advanced Study of Zhongshan University. References [1] M. Asaad, A note on minimal subgroups of 'nite groups, Comm. Algebra 24 (1996) 2771–2776. [2] M. Assad, P. Csorgo, The in4uence of minimal subgroups on the structure of 'nite groups, Arch. Math. 72 (1999) 401–404. [3] M. Asaad, M. Ramadan, A. Shaalan, In4uence of -quasinormality on maximal subgroups of Sylow subgroups of Fitting subgroup of a 'nite group, Arch. Math. 56 (1991) 521–527. [4] A. Ballester-Bolinches, M.C. Pedraza Aguilera, On minimal subgroups of 'nite groups, Acta Math. Hungar 73 (1996) 335–342. [5] A. Ballester-Bolinches, Y. Wang, Finite groups with c-normal minimal subgroups, J. Pure Appl. Algebra 153 (2000) 121–127. [6] J. Buckley, Finite groups whose minimal subgroups are normal, Math. Z. 116 (1970) 15–17. [7] D. Gorenstein, Finite Groups, Chelsea, New York, 1968. [8] B. Huppert, Endliche Gruppen I, Springer, Berlin, 1968. [9] D. Li, X. Guo, The in4uence of c-normality of subgroups on the structure of 'nite groups, J. Pure Appl. Algebra 150 (2000) 53–60. [10] A. Shaalan, A. Ballester-Bolinches, M.C. Pedraza Aguilera, The in4uence of -quasinormality of some subgroups, Acta. Math. Hungar. 56 (1990) 287–293. [11] Y. Wang, c-normality of groups and its properties, J. Algebra 180 (1996) 954–965. [12] Y. Wang, The in4uence of minimal subgroups on the structure of 'nite groups, Acta. Math. Sinica (English series) 16 (2000) 63–70. [13] A. Yokoyama, Finite soluble groups whose F-hypercenter contains all minimal subgroups, Arch. Math. 26 (1975) 123–130. [14] A. Yokoyama, Finite soluble groups whose F-hypercenter contains all minimal subgroups II, Arch. Math. 27 (1976) 572–576.
www.elsevier.com/locate/jpaa
Some notes on the minimal subgroups of Fitting subgroups of 'nite groups a Department
b Department
Li Yangminga; b; ∗
of Mathematics, Zhongshan University, Guangzhou 510275, China of Mathematics, Guangdong College of Education, Guangzhou 510303, China
Received 5 January 2001; received in revised form 4 May 2001 Communicated by G.M. Kelly
Abstract Let G be a 'nite group. The question of how the properties of its minimal subgroups in4uence the structure of G is of considerable interest for some scholars. In this paper, we use the c-normal condition on minimal subgroups of Fitting subgroups to characterize the structure of G through c 2002 Elsevier Science B.V. All the theory of formations, generalizing some known results. rights reserved. MSC: 20D10; 20D30
1. Introduction and preliminaries All groups considered in this note will be 'nite. We use conventional notions and notations, as in [8]. Fix a group G. A minimal subgroup of G is a subgroup of prime order. How minimal subgroups can be embedded in G is a question of particular interest in studying the structure of G (see [1– 6,10 –14]). Ballester-Bolinches and Wang in [5] study the in4uence of c-normality of the minimal subgroups on the structure of G through the theory of formations and obtain many new results which make some classical and recent results as particular cases. In this note, we go further by studying the in4uence of the minimal subgroups of Fitting subgroups of G, improving some results of [5,12]. ∗
Corresponding author. Department of Mathematics, Guangdong College of Education, Guangzhou 510303, China. E-mail address: [email protected] (Li Yangming).
c 2002 Elsevier Science B.V. All rights reserved. 0022-4049/02/$ - see front matter PII: S 0 0 2 2 - 4 0 4 9 ( 0 1 ) 0 0 1 2 6 - 8
290
Li Yangming / Journal of Pure and Applied Algebra 171 (2002) 289–294
Let F be a class of groups, we call F a formation provided that (i) if G ∈ F and H / G, then G=H ∈ F, and (ii) if G=M and G=N are in F, then G=(M ∩ N ) is in F for normal subgroups M; N of G. A formation F is said to be saturated if G=(G) ∈ F implies that G ∈ F (see [8, Chapter VI]). Throughout this paper U; N will denote the class of all supersolvable groups and the class of all nilpotent groups, respectively. Clearly, U and N are formations. Since a group G is supersolvable iH G=(G) is supersolvable (see [8, p. 713, Satz 8.6]), it follows that U is saturated. N is saturated too (see [8, p. 220, Satz 3.5]). Let F be a saturated formation. A normal subgroup N of a group G is said to be F-hypercentral in G provided N has a chain of subgroups 1 = N0 / N1 / · · · / Nr = N such that Ni+1 =Ni is an F-central chief factor of G. The product of all F-hypercentral subgroups of G is again an F-hypercentral subgroup of G. It is denoted by ZF (G) and called the F-hypercenter of G. For a formation F, each group has a smallest normal subgroup N such that G=N is in F. This uniquely determined normal subgroup of G is called the F-residual subgroup of G and is denoted by G F . For the formation N, some authors prefer to write K∞ (G) as G N and Z∞ (G) as ZN (G). (see [8, Satz III 2.5(d)]). We say, following Wang [11] that a subgroup H of a group G is c-normal in G if there exists a normal subgroup N of G such that G = HN and H ∩ N 6 HG = CorG (H ). Let x be an element of G, we say that x is c-normal in G if x is c-normal in G. First, we give some known results. Lemma 1. Let G be a group. Then (1) If H is normal in G; then H is c-normal in G. (2) If H is c-normal in G; H 6 K 6 G; then H is c-normal in K. (3) Let K / G and K 6 H . Then H is c-normal in G if and only if H=K is c-normal in G=K. (4) Let be a set of primes; H is a normal -subgroup of G and T a -subgroup of G. If T is c-normal in G; then TH=H is c-normal in G=H . (5) If P be a minimal normal p-subgroup of G and x ∈ P is c-normal in G; then P = x. Proof. Conditions (1) – (3) can be found in [11, Lemma 2:1], (4) in [9, Lemma 2:4], and (5) in [12, Lemma 2:2(1)]. Lemma 2. Let P be a normal p-subgroup of G and H / G. Then F(HP) = F(H )P. Proof. See [9, p. 522, Lemma 3:1]. Lemma 3. Let F be a saturated formation containing N. Suppose that G is a group with a normal subgroup H such that G=H ∈ F. Then G ∈ F if and only if every
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element of order 4 of H is c-normal in G and x lies in the F-hypercenter ZF (G) of G for every element x of prime order of H . Proof. See [5, Theorem 3:1]. Lemma 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 c-normal in G; then G ∈ F. Proof. See [5, Theorem 3:4]. 2. Results Theorem 1. Let F be a saturated formation containing N 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 every element of order 4 of F(H ) is c-normal in G and x lies in the F-hypercenter ZF (G) of G for every element x of prime order of F(H ). Proof. (a) ⇒ (b) If G ∈ F, then (b) is true with H = 1. (b) ⇒ (a) Suppose that the results are false and let G be a counterexample of minimal order. Obviously, H = 1, then F(H ) = 1 by the hypothesis of the solvability of H . Let p be a prime dividing |F(H )| and let P ∈ Sylp (F(H )). Clearly, P / G. Once we know that G=P inherits the Hypothesis (b), we conclude G=P ∈ F by the minimality of |G|. Applying, Lemma 3, 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. Since F(H=P) = F(H )=P by Lemma 2, we have to show that the cyclic subgroup of order 4 of F(H )=P are c-normal in G=P and every subgroup of prime order of F(H )=P lies in the F-hypercenter ZF (G=P) of G=P. If p = 2, then there exists no element of order 4 in F(H )=P. Hence suppose p = 2. Let X=P be a cyclic subgroup of F(H )=P of order 4 . By the Schur– Zassenhaus Theorem [7, p. 221, Theorem 2:1], X = xP where x is an element of order 4 of F(H ). Since x is c-normal in G by hypothesis, X=P is c-normal in G=P by Lemma 1(4). Since ZF (G)P=P 6 ZF (G=P), it is easy to see that every subgroup of prime order of F(H )=P lies in the F-hypercenter ZF (G=P) by the hypothesis on the element of prime order of F(H ). These 'nish the proof. Remark 1. An equivalent form of Theorem 1 is as follows (comparing with [5, Theorem 3:1]): Theorem 1 . Let F be a saturated formation containing N and let G be a group. Equivalent are (a) G ∈ F.
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(b) G F is solvable and every element of order 4 of F(G F ) is c-normal in G and x lies in the F-hypercenter ZF (G) of G for every element x of prime order of F(G F ). Remark 2. If F = N, then Theorem 1 coincides with [12, Theorem 3:1]. Theorem 2. Let F be a saturated formation containing N and let G be a group with a normal solvable subgroup H in G such that G=H ∈ F. Then G ∈ F under either of the following: (a) G is 2-nilpotent and x lies in the F-hypercenter ZF (G) of G for every element x of prime order of F(H ). (b) The Sylow 2-subgroups of H are abelian and x lies in the F-hypercenter ZF (G) of G for every element x of prime order of F(H ). Proof. (a) Suppose G is 2-nilpotent, then H is also 2-nilpotent. Denote K is the 2-complement of H , then K / G. Since (G=K)=(H=K) ∼ = G=K ∈ F, and F(H=K) = H=K is a 2-subgroup, it follows that G=K ∈ F by the hypotheses on F and induction on |G|. Also, F(K) 6 F(H ). So we may substitute H by K which is of odd order. Then, the result now follows applying Theorem 1. (b) Let P be an abelian Sylow 2-subgroup of H and put P1 = F(H ) ∩ P. For any x in 1 (P1 ), consider the minimal 2-subgroup P2 of G contained in xG = xg | g ∈ G which is the minimal normal subgroup of G containing x. Then P2 6 F(H ). By the hypothesis of F(H ) and Lemma 1(5), we have that P2 = xg for some g ∈ G. Hence xg lies in the center of G, so does x. It follows that 1 (P1 ) ⊆ Z(G). By [7, p. 178, Theorem 2:4], every element of odd order centralizes P1 , thus P1 is in the center of G since P is abelian. Since G=P1 inherits the hypothesis, G=P1 ∈ F. Thus G ∈ F by the proof of our Theorem 1. Remark 3. Theorem 2 is analogous to [4, Theorems 3 and 4]. Theorem 3. Let F be a saturated formation containing 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 order 4 of the Fitting subgroup F(H ) are c-normal in G. Proof. (a) ⇒ (b) If G ∈ F, then (b) is true with H = 1. (b) ⇒ (a) Suppose that the results are false and let G be a counterexample of minimal order. Obviously, H = 1, then F(H ) = 1 by the hypothesis of the solvability of H . Let p be a prime dividing |F(H )| and let P ∈ Sylp (F(H )). Clearly, P / G. Once we know that G=P inherits the Hypothesis (b), we conclude G=P ∈ F by the minimality of |G|. Applying, Lemma 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. Since F(H=P) = F(H )=P by
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Lemma 2, we have to show that the subgroup of prime order or 4 of F(H )=P are c-normal in G=P. Let X=P be a subgroup of F(H )=P of prime order q (of order 2 or 4 if q = 2). Obviously, q = p. By the Schur–Zassenhaus Theorem [7, p. 221, Theorem 2:1], X = xP where x is an element of prime order or 4 of F(H ). Since x is c-normal in G by hypothesis, thus X=P is c-normal in G=P by Lemma 4(4). These 'nish the proof. Remark 4. An equivalent form of Theorem 3 is as follows (comparing with [5, Theorem 3:4]): Theorem 3 . Let F be a saturated formation containing U and let G be a group. Equivalent are (a) G ∈ F. (b) G F is solvable and every element of prime order or order 4 of F(G F ) is c-normal in G. Remark 5. Theorem 3 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. Remark 6. Theorem 3 is not true if we omit the solvability of H . Set G = H × K, where H = SL(2; 5) and K ∈ U. Then |F(H )| = 2 and G=H ∼ = K ∈ U, but G ∈ U. Remark 7. If F = U, then Theorem 3 coincides with [12, Theorem 3:2]. Theorem 4. Let F be a saturated formation containing U and let G be a group with a normal solvable subgroup H in G such that G=H ∈ F. Then G ∈ F under either of the following: (a) G is 2-nilpotent and every subgroup of odd prime order of F(H ) is c-normal in G. (b) The Sylow 2-subgroups of G are abelian and every subgroup of F(H ) of prime order is c-normal in G. Proof. (a) Suppose G is 2-nilpotent, then H is also 2-nilpotent. Denote K is the 2-complement of H , then K / G. Since (G=K)=(H=K) ∼ = G=K ∈ F, and F(H=K) = H=K is a 2-subgroup, it follows that G=K ∈ F by the hypotheses on F and induction on |G|. Also, F(K) 6 F(H ). So we may substitute H by K which is of odd order. Then, the result now follows applying Theorem 3. (b) Let P be an abelian Sylow 2-subgroup of G and put P1 = F(H ) ∩ P. For any x in 1 (P1 ), consider the minimal 2-subgroup P2 of G contained in xG = xg | g ∈ G which is the minimal normal subgroup of G containing x. Then P2 6 F(H ). By the hypothesis of F(H ) and Lemma 1(5), we have that P2 = xg for some g ∈ G. Hence xg lies in the center of G, so does x. It follows that 1 (P1 ) ⊆ Z(G). By [7, p. 178, Theorem 2:4], every element of odd order centralizes P1 , thus P1 is in the center of
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G since P is abelian. Since G=P1 inherits the hypothesis, G=P1 ∈ F. Thus G ∈ F by the proof of our Theorem 3. Acknowledgements The author is very much indebted to Professor Wang Yanming for his encouragement and advice. The research is supported by the Foundation of advanced Study of Zhongshan University. References [1] M. Asaad, A note on minimal subgroups of 'nite groups, Comm. Algebra 24 (1996) 2771–2776. [2] M. Assad, P. Csorgo, The in4uence of minimal subgroups on the structure of 'nite groups, Arch. Math. 72 (1999) 401–404. [3] M. Asaad, M. Ramadan, A. Shaalan, In4uence of -quasinormality on maximal subgroups of Sylow subgroups of Fitting subgroup of a 'nite group, Arch. Math. 56 (1991) 521–527. [4] A. Ballester-Bolinches, M.C. Pedraza Aguilera, On minimal subgroups of 'nite groups, Acta Math. Hungar 73 (1996) 335–342. [5] A. Ballester-Bolinches, Y. Wang, Finite groups with c-normal minimal subgroups, J. Pure Appl. Algebra 153 (2000) 121–127. [6] J. Buckley, Finite groups whose minimal subgroups are normal, Math. Z. 116 (1970) 15–17. [7] D. Gorenstein, Finite Groups, Chelsea, New York, 1968. [8] B. Huppert, Endliche Gruppen I, Springer, Berlin, 1968. [9] D. Li, X. Guo, The in4uence of c-normality of subgroups on the structure of 'nite groups, J. Pure Appl. Algebra 150 (2000) 53–60. [10] A. Shaalan, A. Ballester-Bolinches, M.C. Pedraza Aguilera, The in4uence of -quasinormality of some subgroups, Acta. Math. Hungar. 56 (1990) 287–293. [11] Y. Wang, c-normality of groups and its properties, J. Algebra 180 (1996) 954–965. [12] Y. Wang, The in4uence of minimal subgroups on the structure of 'nite groups, Acta. Math. Sinica (English series) 16 (2000) 63–70. [13] A. Yokoyama, Finite soluble groups whose F-hypercenter contains all minimal subgroups, Arch. Math. 26 (1975) 123–130. [14] A. Yokoyama, Finite soluble groups whose F-hypercenter contains all minimal subgroups II, Arch. Math. 27 (1976) 572–576.