Finite Soluble Groups in Which the Normalizer of Every Non-Normal Cyclic Subgroup Is Maximal

J. Group Theory 17 (2014), 671–687 DOI 10.1515/jgt-2014-0004 © de Gruyter 2014

Finite soluble groups in which the normalizer of every non-normal cyclic subgroup is maximal

Jianji Cao and Xiuyun Guo Communicated by Evgenii I. Khukhro

Abstract. In this paper a classiﬁcation is given for ﬁnite soluble groups in which the normalizer of every non-normal cyclic subgroup is a maximal subgroup.

1 Introduction

All groups considered in this paper are ﬁnite. It is well known that the normality of subgroups plays an important part in research in group theory. So it is reasonable to investigate the structure of a group by using normalizers of certain types of subgroups. For example, Bianchi, Gillio Berta Mauri and Hauck in [2] proved that a group is nilpotent if and only if the normalizer of every Sylow subgroup is nilpotent. Ballester-Bolinches and Shemetkov in [1] gave a beautiful criterion for nilpotent groups: a group is nilpotent if and only if the normalizer of every Sylow p-subgroup is p-nilpotent for every prime p. Zhang in [11] investigated the structure of groups using Sylow numbers (that is, the index of the normalizer of a Sylow subgroup). On the other hand, there is con- siderable research in the literature concerning the relationship between the structure of p-groups and the normalizers of certain kinds of subgroups (see [6, 8, 9]). For example, Parmeggiani investigates the p-groups with many subgroups that are 2-subnormal [9]. Ormerod investigates the p-groups in which every cyclic subgroup is 2-subnormal [8]. Inspired by the above research, we are interested in the class of groups in which the normalizer of every non-normal cyclic subgroup is a maximal subgroup. We note that Mann in [7] investigated non-soluble groups in which the normalizer of every non-normal subgroup is maximal and he also gave a classiﬁcation for this kind of group. Although the class of groups in which the normalizer of every non-normal cyclic subgroup is maximal is different from the class of groups in which the normalizer of every non-normal subgroup is maximal (see the following

The research presented here was partially supported by the National Natural Science Foundation of China (11371237) and a grant of “The First-Class Discipline of Universities in Shanghai”. The corresponding author is Xiuyun Guo. 672 J. Cao and X. Guo

Examples 2.1 and 2.2), we only investigate soluble groups in which the normalizer of every non-normal cyclic subgroup is a maximal subgroup in this paper and then give a classiﬁcation for this kind of groups.

2 Preliminaries

In this section we list some basic properties of groups in which the normalizer of every non-normal cyclic subgroup is a maximal subgroup and other useful results for us. For convenience, we call a group an NCM-group if the normalizer of every non-normal cyclic subgroup is a maximal subgroup. We begin with some examples which illustrate that there exist NCM-groups and that there also exists a non-normal subgroup in these NCM-groups such that its normalizer is not maximal.

3 Example 2.1. Let p be a prime and let Mi be a non-abelian group of order p with exponent p for i 1; 2; : : : ; k and k > 2. Then the central product D P M1 M2 M D k is an NCM-group and there exists a non-normal subgroup H in P such that the normalizer NP .H / is not maximal in P .

Proof. Let a be a non-normal cyclic subgroup of P . Then there exists an ele- h i ment b P such that Œa; b c 1. Since P is an extra special p-group, we 2 D ¤ have Z.P / c , and therefore a; b Mi . Thus we get P a; b CP . a; b / D h i h i ' D h i h i by [4, Lemma 5.4.6]. It follows that NP . a / is maximal in P , and therefore P is h i an NCM-group. Let Mi ai ; bi for i 1; 2; : : : ; k, and let H a1; a2 with a1 µ M1, D h i D b D h i b h i a2 µ M2. Thus H a1 a2 . Since a1 1 a1c H and a2 2 a2c H, h i D h ih i D … D … we see b1 and b2 are not in NP .H /. Noting that both b1 and b2 are not in ˆ.P /, we see NP .H / is not maximal in P .

Example 2.2. Let G a; b; c a3 b3 c4 1; ac ab2; bc a2b2; Œa; b 1 : D h j D D D D D D i Then G is an NCM-group and there exists a non-normal subgroup H in G such that NG.H / is not maximal in G.

Proof. Since G . a b / Ì c , we see that G has no non-trivial cyclic nor- D h i h i h i 2 mal subgroup and that G is an NCM-group. Take H a c . Since b NG.H / D h ih 2 i … and c NG.H /, we see NG.H / < M . a b / Ì c < G. So NG.H / is … D h i h i h i not maximal in G. Finite soluble groups 673

Lemma 2.3. Let q be a prime dividing the order of a group G. If G is an NCM- group and A is a non-normal cyclic q-subgroup of G, then there exists a normal Hall q0-subgroup T in CG.A/ and every subgroup of T is normal in NG.A/. Par- ticularly, if q is the smallest prime dividing the order of G, then NG.A/ T Ì Q D with T an abelian q0-subgroup and Q a Sylow q-subgroup of NG.A/.

Proof. If CG.A/ is a q-subgroup, then the lemma is true. Assume that CG.A/ is not a q-subgroup and that B is a cyclic p-subgroup of CG.A/ with p q a prime. ¤ Since A is a characteristic subgroup of AB, we see AB µ G and A E NG.AB/. It follows from the maximality of NG.A/ that NG.A/ NG.AB/. By the same D reason, we have that B E NG.AB/ and therefore NG.B/ > NG.AB/ NG.A/. D It follows that the Sylow p-subgroup in CG.A/ is normal in NG.A/ and therefore the Hall q0-subgroup T in CG.A/ is normal in NG.A/. By the above proof, we see that every subgroup in T is normal in NG.A/. In particular, if q is the smallest prime dividing the order of G, then the Hall q0-subgroup of NG.A/ is contained in CG.A/ since NG.A/=CG.A/ is a q-subgroup. Hence NG.A/ T Ì Q with T an abelian q -subgroup and Q a Sylow D 0 q-subgroup of NG.A/.

Lemma 2.4. Let N be a normal subgroup of a group G. If G is an NCM-group, then G=N is also an NCM-group.

Proof. Let x N=N µ G=N . Then x µ G, therefore the normalizer NG. x / h i h i h i is a maximal subgroup in G. It follows from NG. x /N=N 6 N . x N=N / h i G=N h i that N . x N=N / is a maximal subgroup in G=N . G=N h i Lemma 2.5 ([3, Theorem A.15.2]). Let G be a primitive group with stabilizer M . Then exactly one of the following three statements holds: (1) G has a unique minimal normal subgroup N , this subgroup N is self-central- izing (in particular, abelian), and N is complemented by M in G. (2) G has a unique minimal normal subgroup N , this N is non-abelian, and N is supplemented by M in G.

(3) G has exactly two minimal normal subgroups N and N , and each of them is complemented by M in G. Also one has CG.N / N , CG.N / N , and D D N N NN M . Moreover, if V < G and VN VN G, then one ' ' \ D D has V N V N 1. \ D \ D Recall that an automorphism of a group is called a power automorphism if it leaves every subgroup invariant. 674 J. Cao and X. Guo

Lemma 2.6 ([10, 13.4.3]). Let ˛ be a power automorphism of an abelian group A. If A is a p-group, then there is a positive integer l such that a˛ al for all a A. D 2 If ˛ is nontrivial and has order prime to p, then ˛ is ﬁxed-point-free.

Lemma 2.7 ([5, Theorem 3.8.2]). Let G be a p-group. If G has a unique cyclic subgroup of order p, then (1) G is a cyclic group for p > 2. (2) G is either a cyclic group or a generalized quaternion group for p 2. D Lemma 2.8 ([5, Theorem 6.14.4]). Let G be a soluble group with all Sylow subgroups abelian, and let D be a system normalizer of G. Then D is the complement of G0 in G.

3 Main results

In this section the main results in this paper are given.

Lemma 3.1. Let G be a soluble NCM-group. If there exists no non-trivial cyclic normal subgroup in G, then there is a maximal subgroup M in G such that coreG.M / 1, and therefore G has a unique minimal normal subgroup. D Proof. Suppose that there exists no non-trivial cyclic normal subgroup in G and coreG.M / 1 for every maximal subgroup M in G. Then G is a non-nilpotent ¤ group. Let q be a minimal prime dividing the order of G and let Q be a Sylow q-subgroup of G. Then: (1) One has Oq .G/ 1. 0 ¤ In fact, if Oq .G/ 1, then F .G/ Oq.G/. Let y be a cyclic p-subgroup 0 D D h i of G with p q. Then NG. y / is maximal in G and coreG.NG. y // 1. Thus ¤ h i h i ¤ there is a minimal normal subgroup N of G such that N 6 NG. y / and N is an h i elementary abelian q-group. Let x 6 N Z.Q/ be a cyclic subgroup of order q. h i \ Then, by Lemma 2.3, NG. x / Tx Ì Q with Tx a Hall q -subgroup of NG. x /. h i D 0 h i Noting that Oq.G/ 6 Q, we see ŒTx;Oq.G/ 6 Tx Oq.G/ 1, and therefore \ D Tx 6 CG.Oq.G// CG.F .G// 6 Oq.G/, in contradiction to that 1 y Tx D ¤ 2 is a p-element. (2) Let NG. x / Tx Ì Qx with Qx a Sylow q-subgroup of NG. x / and Tx h i D h i a Hall q -subgroup of NG. x / for x Q. If Tx 1, then NG. x / E G if and 0 h i 2 ¤ h i only if coreG.Tx/ 1. ¤ In fact, if NG. x / E G, then Tx E G. On the other hand, if coreG.Tx/ 1, h i ¤ then g g g ŒNG. x / Tx Ì Qx > coreG.Tx/ h i D Finite soluble groups 675

for any g G. Now let 1 y coreG.Tx/. Then y µ G by hypothesis. Thus, 2 ¤ 2 h i by Lemma 2.3, g ŒNG. x / NG. y / NG. x /: h i D h i D h i So NG. x / E G. h i (3) NG. x / E G for any 1 x Q. h i ¤ 2 In fact, by (1), there is a minimal normal subgroup K in G such that K is an elementary abelian p-group with p q. If there exists an element x Q such ¤ 2 that NG. x / µ G, then, by (2), G KNG. x /. Thus by Lemma 2.3, we have h i D h i G K Ì .Tx Ì Q/ with Tx a Hall q -subgroup of NG. x /. On the other hand, D 0 h i it follows from coreG.NG. x // 1 and (2) that there exists a minimal normal h i ¤ subgroup N of G such that N 6 Q. Noting that G is a q-nilpotent group, we i see G CG.N / q by Frobenius’s criterion for q-nilpotent. So, for any ele- j W j D ment y N Z.Q/, we have y E G, a contradiction. Hence NG. x / E G 2 \ h i h i for any 1 x Q. ¤ 2 (4) . NG. x / ; G NG. x / / 1 for any 1 x Q and Q is a Dedekind j h i j j W h i j D ¤ 2 group. In fact, let NG. x / Tx Ì Qx with Qx a Sylow q-subgroup of NG. x / and h i D h i Tx a Hall q -subgroup of NG. x /. If Tx 1, then NG. x / Qx is maximal 0 h i D h i D in G, and therefore the conclusion is true. Thus we only need to consider Tx 1. ¤ It is clear that we may assume that Qx 6 Q and G NG. x / p is a prime. j W h i j D Let Px be a Sylow p-subgroup of NG. x / and let P be a Sylow p-subgroup of G h i with Px 6 P . If p .Tx/, then Px E G by Lemma 2.3. Thus there exists an el- 2 ement h such that 1 h Z.P / Px. By Lemma 2.3, we have that h E G, ¤ 2 \ h i a contradiction. If p q, then we may assume P Q. Take a Z.Q/. Then D D 2 Q 6 NG. a / and therefore G NG. a /NG. x /. Let NG. a / Ta Ì Q. The h i D h i h i h i D normality of NG. a / implies Ta E G and therefore we obtain Ta 6 Tx. Since h i G NG. a / r is a prime and r q, we see .r; Ta / 1 by the above proof, j W h i j D ¤ j j D and therefore . Tx Ta ; Ta / 1. Noting that both Tx and Ta are normal in G j W j j j D and Tx is abelian, we see that the complement of Ta in Tx is a cyclic normal subgroup of G, another contradiction. Hence . G NG. x / ; NG. x / / 1, which j W h i j j h i j D also proves that Q is a Dedekind group. (5) There exists a maximal subgroup M in G such that coreG.M / 1. D In fact, by (3) we can let G NG. x / p for 1 x Q and let H be a Sy- j W h i j D ¤ 2 low p-subgroup of G. By (4), NG. x / is a Hall p -subgroup of G with p q and h i 0 ¤ H is a cyclic group of order p. Also by (3), we have NG. x / NG.H / E NG.H / h i \ and NG. x / NG.H / is a Hall p -subgroup of NG.H /. Hence h i \ 0 NG.H / H .NG. x / NG.H //: D h i \ It follows from Lemma 2.3 that every cyclic q -subgroup of NG. x / NG.H / is 0 h i \ normal in G. Thus NG. x / NG.H / is a q-subgroup. Noting that Q is a Dedekind h i \ 676 J. Cao and X. Guo group, we see that NG. y / NG.H /; Q G h i D h i D for any y NG. x / NG.H /. So we get NG. x / NG.H / 1. Consequently, 2 h i \ h i \ D NG.H / H is a maximal subgroup of G and coreG.H / 1. By Lemma 2.5, D D there is a unique minimal normal subgroup in G. The proof is now complete.

Lemma 3.2. Let q be the smallest prime dividing the order of a group G. If G is a soluble NCM-group, then G is either q-closed or q-nilpotent.

Proof. Suppose that the result is not true and let G be a counterexample of minimal order. Let Q be a Sylow q-subgroup of G. Then: (1) There is no non-trivial cyclic normal q-subgroup in G. If there is an element x G of order q such that x E G, then 2 h i x 6 Z.G/ Q K: h i \ D By Lemma 2.4, G=K is an NCM-group. The minimality of G implies that G=K is q-closed or q-nilpotent. It follows that G is q-closed or q-nilpotent, a contradiction. (2) There is no non-trivial cyclic normal q0-subgroup in G. In fact, suppose that there is a non-trivial cyclic normal q0-subgroup in G and that N is the product of all cyclic normal q0-subgroups in G. By Lemma 2.4, G=N is an NCM-group, and therefore G=N is q-closed or q-nilpotent. If G=N is q-nilpotent, then G is also q-nilpotent, a contradiction. So we may assume that G=N is q-closed and that QN is normal in G. By Lemma 2.3, NG. x / Tx Ì Qx with Tx a Hall q -subgroup of NG. x / h i D 0 h i and Qx a Sylow q-subgroup of NG. x / for every 1 x Q. If x E Q, then h i ¤ 2 h i Qx Q. We conclude that N 6 Tx. Otherwise, the maximality of NG. x / im- D h i plies that G NNG. x / is q-nilpotent, a contradiction. So we get x CG.N /. D h i 2 If x µ Q, then Tx is a Hall q -subgroup of G. Thus N 6 Tx, and therefore h i 0 x CG.N /. So Q 6 CG.N /. Therefore Q is normal in G, a contradiction. 2 (3) Final contradiction. By (1) and (2), we see that there is no non-trivial cyclic normal subgroup in G. Lemma 3.1 implies that there must be a maximal subgroup M of G such that coreG.M / 1 and G has a unique minimal normal subgroup K. So G MK, D D M K 1 and F .G/ K. \ D D Let K be an elementary abelian p-group for some prime p. Then M must be a p -group. Otherwise, there exists an element y M such that y is a p-group. If 0 2 h i K 6 NG. y /, then we have y CG.K/, in contradiction to CG.F .G// F .G/. h i 2 D If K NG. y /, then the maximality of NG. y / implies that G NG. y /K. — h i h i D h i So K Ì y is a normal p-subgroup of G, in contradiction to F .G/ K. h i D Finite soluble groups 677

So if p q, then G is q-closed, a contradiction. Hence we assume p q. D ¤ Since G is not q-nilpotent, there exists a q -element x M g for some element 0 2 g G such that x Oq .G/. If Oq .G/ 6 NG. x /, then, since K Oq .G/, we 2 … 0 0 h i Ä 0 have ŒK; x 6 K x 1 and therefore x CG.K/ CG.F .G// K, a con- \ h i D 2 D D tradiction. If Oq .G/ NG. x /, then we get G Oq .G/NG. x / and therefore 0 — h i D 0 h i Oq .G/ Ì x is normal q -subgroup of G, a contradiction. These contradictions 0 h i 0 show that the result is true.

Theorem 3.3. Let G be a nilpotent group. Then G is an NCM-group if and only if all Sylow subgroups in G are NCM-groups and there is at most one Sylow subgroup in G is not a Dedekind group. Proof. It is easy to see that we only need to prove necessity. Let

G P1 P2 Ps D with Pi a Sylow pi -subgroup. If there exist elements xi Pi and xj Pj such 2 2 that xi µ Pi and xj µ Pj , then it follows from h i h i NG. xi xj / NG. xi xj / NG. xi / NG. xj / h ih i D h i D h i \ h i that NG. xi xj / NG. xi / NG. xj / h ih i D h i D h i and therefore Pi Pj 6 NG. xi /, in contradiction to xi µ Pi and xj µ Pj . h i h i h i Now suppose P Syl .G/ and P is not a Dedekind group. If x µ P , then 2 p h i the maximality of NG. x / implies that NG. x / E G and G NG. x / p. h i h i j W h i j D Hence P NP . x / p and P is an NCM-group. j W h i j D Lemma 3.4. Let q be the smallest prime dividing the order of a group G and let Q be a Sylow q-subgroup in G. If G is a soluble non-nilpotent NCM-group and G has a normal q-complement T , then:

(1) T 6 CG.x/ if x E G or x µ Q for x Q. h i h i 2 (2) There is a prime p such that G NG. y / is a power of p for every y Q j W h i j 2 with y E Q but y µ G. h i h i (3) CT .y/ CT .Q/ for every y Q with y E Q but y µ G. D 2 h i h i (4) P is normal in G if P is a Sylow p-subgroup of T .

(5) P is an abelian group and CP .Q/ is in the center of G. (6) If there is an element t T with t E T but t µ G, then ŒP; Q is a mini- 2 h i h i mal normal subgroup in G.

Proof. (1) It is easy to ﬁnd that NG. x /=CG.x/ is a q-group for every x Q by h i 2 the minimality of q. It follows that (1) is true. 678 J. Cao and X. Guo

(2) Suppose the result is not true. Then there are two elements yi Q with 2 yi E Q but yi µ G for i 1; 2 such that h i h i D l m G NG. y1 / p1 and G NG. y2 / p2 j W h i j D j W h i j D with primes p1 and p2 and p1 p2. Let P1 be a Sylow p1-subgroup of T ¤ such that P1 6 NT . y2 / and let P2 be a Sylow p2-subgroup of T such that h i P2 6 NT . y1 /. Then we get Œy1y2;P1 1 and Œy1y2;P2 1. If T is a nilpo- h i ¤ ¤ tent group, then p1p2 divides G NG. y1y2 / , in contradiction to the fact that j W h i j NG. y1y2 / is a maximal subgroup. So T is a non-nilpotent group. By Lemma 2.3, h i NG. yi / CT .yi / Ì Q with CT .yi / an abelian Hall q -subgroup of NG. yi / h i D 0 h i for i 1; 2, and hence we have NG. y2 / 6 NG.P1/ and NG. y1 / 6 NG.P2/. D h i h i If NT .P1/ CT .P1/, then T is p1-nilpotent. If NT .P1/ CT .P1/, then there D ¤ exists an element t NT .P1/ CT .P1/. Since CT .y2/ 6 CT .P1/, we conclude 2 n that t NG. y2 /. The maximality of NG. y2 / implies that P1 E G. Thus T is … h i h i p1-nilpotent or P1 E T . Similarly, T is p2-nilpotent or P2 E T . If P1 and P2 are normal in T , then, since the Hall p1; p2 -subgroup in T is contained in ¹ º0 the center of T , it is clear that T is nilpotent. Similarly, T is also nilpotent if T is p1-nilpotent and p2-nilpotent. So we may assume that P1 is normal in T and that T is p2-nilpotent. In this case we may assume T H .P1 Ì P2/ with D H an abelian Hall p1; p2 -subgroup of T . Noting that every element of Q in- ¹ º0 duces a power automorphism on P1 and P2 by Lemma 2.3, we see that y1 induces a non-trivial p1 -automorphism on P1 since P1 CT .y1/. By Lemma 2.6, y1 can 0 — induce a ﬁxed-point-free automorphism on P1. Thus CP .y1/ 1, and therefore 1 D CT .y1/ is a Hall p10-subgroup of T . g Suppose that there exists an element g P1 such that Œy1y2;P2 1. By the 2 D above arguments, y1 induces a ﬁxed-point-free automorphism on g . So we may h i assume gy1 gj with .j; g / 1 and j 1 .mod g /. On the other hand, D jh ij D 6Á jh ij since P2 µ T , there exists d 6 P2 such that d µ T . Similarly, y2 induces h i h i a power automorphism on d by conjugation. We may assume d y2 d k with h ig g D .k; d / 1. Since NG. y1 / is maximal and d µ T , we see jh ij D h i h i g g g g g g NG. y1 / NG. y1 / .H P2 / Ì Q NG. d / h i D h i D D h i j 1 g g k by Lemma 2.3. Noting that g NT . d / NT . .d / /, we see … h i D h i j y1y2 g d g .d g /y1y2 .d y1y2 /g .d k/ D D D j 1 j 1 .d gg /k ..d g /k/g .d g /k d g ; D D … h i D h i g a contradiction. Hence Œy1y2;P2 1 for any g G, in contradiction to that ¤ 2 NG. y1y2 / is a maximal subgroup. So (2) is true. h i Finite soluble groups 679

(3) If not, then there are two elements yi Q with yi E Q but yi µ G for 2 h i h i i 1; 2 such that CT .y1/ CT .y2/. By Lemma 2.3, NG. yi / CT .yi / Ì Q D ¤ h i D with CT .yi / an abelian Hall q -subgroup of NG. yi / for i 1; 2. Then: 0 h i D (i) T is a non-nilpotent group. In fact, if T is a nilpotent group, then it follows from CT .y1/ CT .y2/ and (2) ¤ that CP .y1/ CP .y2/ if P is a Sylow p-subgroup of T . Noting that y1 in- ¤ duces a p0-power automorphism on CP .y2/ by Lemma 2.3 and y1 induces an identity automorphism on CP .y1/, we see CP .y2/ 6 CP .y1/ by Lemma 2.6 if CP .y1/ CP .y2/ 1. Similarly, CP .y1/ 6 CP .y2/ if CP .y1/ CP .y2/ 1. \ ¤ \ ¤ Therefore CP .y1/ CP .y2/, a contradiction. So CP .y1/ CP .y2/ 1 and P D \ D is an elementary abelian group, and furthermore T is an abelian group. By Lem- ma 2.3 again, Q normalizes every subgroup of CP .y1/ and CP .y2/. Thus every subgroup of CP .y1/ and CP .y2/ is normal in G. The maximality of NG. y1 / h i and NG. y2 / imply that CP .y1/ 1 and CP .y2/ 1, and therefore we con- h i ¤ ¤ 2 clude that P CP .y1/ CP .y2/ is an elementary abelian group of order p . Let D CP .y1/ a and CP .y2/ b . It is clear that Œy1y2; a 1 and Œy1y2; b 1. D h i D h i ¤ ¤ If CP .y1y2/ 1, then the maximality of the normalizer NG. y1y2 / implies that D h i G NG. y1y2 /; a , and therefore P p, a contradiction. Thus there exists D h h m ni i j j D m n an element a b P with .mn; p/ 1 such that a b CT .y1y2/. On the 2 D 2 other hand, m n m n m n m n Œy1y2; a b Œy1; a b Œy1; a b ; y2Œy2; a b : D m n n m m n n n Since 1 Œy1; a b Œy1; b Œy1; a Œy1; a ; b Œy1; b b b and ¤ m n D n m m n D m m2 h i D h i 1 Œy2; a b Œy2; b Œy2; a Œy2; a ; b Œy2; a a a , we see ¤ m n D D 2 h i D h i Œy1y2; a b 1, a contradiction. So T is a non-nilpotent group. ¤ (ii) One has P E T . Assume CT .y1/ M P1 P P , where M is the product of all D l normal Sylow subgroups of T which are contained in CT .y1/, Pk is a Sylow p -subgroup of T for k 1; 2; : : : ; l and P is a Sylow p-subgroup of CT .y1/. k D The maximality of NG. y1 / implies that h i G CT .y1/; CT .y2/; Q CT .y1/; CT .y2/ Q: D h i D h i Thus T 6 CT .y1/; CT .y2/ . By part (2), M 6 CT .y1/ CT .y2/. Since CT .y1/ h i \ and CT .y2/ are abelian groups, we see M 6 Z.T /. Then it follows from (i) that M is not a Hall p0-subgroup of T , and therefore l > 1. If there exists an element t NT .P / CT .P /, then, since CT .y1/ 6 CT .P /, we see t NG. y1 /. 2 k n k k … h i It follows from Lemma 2.3 that NG. y1 / < NG.P /, and therefore P E G, h i k k a contradiction. Hence NT .P / CT .P / for k 1; 2; : : : ; l, and therefore T k D k D has normal pk-complement Tp0k . It is clear that the intersection of all Tp0k for k 1; 2; : : : ; l is equal to M P , which yields that P is normal in T . D 680 J. Cao and X. Guo

(iii) CT .y1/ and CT .y2/ are Carter subgroups of T . If there exists an element t T CT .y1/ such that t normalizes CT .y1/, then 2 n the maximality of NG. y1 / implies that CT .y1/ is normal in G and therefore h i T is nilpotent by (ii), a contradiction. Thus CT .yi / is self-normalizing in T , and furthermore CT .yi / for i 1; 2 are Carter subgroups of T . D (iv) Final contradiction. g By (iii) there exists an element g P such that CT .y1/ CT .y2/ . It is clear g g g 2 D that y2 E Q but y2 µ G. By Lemma 2.3, h i h i g g g g g NG. y2 / NG. y2 / CT .y2/ Ì Q CT .y1/ Ì Q : h i D h i D D g g t If Q 6 NG. y1 /, then there exists an element t CT .y1/ such that Q Q h i 1 2 1 D by Sylow’s theorem. Thus gt NT .Q/ and so gt CT .Q/ 6 CT .y1/ by 2 2 Frobenius’s criterion for q-nilpotency. Hence g CT .y1/, a contradiction. So we g g 2 may assume Q NG. y1 /. Since both Q and NG. y1 / normalize CT .y1/, — h i h i we see that CT .y1/ is normal in G by the maximality of NG. y1 /, in contradic- h i tion to (iii). So (3) is true. (4) It is clear that the result is true if T is a nilpotent group. Now we may assume that T is a non-nilpotent group and that CT .Q/ CP .Q/ P1 P2 Ps D with Pr a Sylow pr -subgroup of T for r 1; 2; : : : ; s. D The maximality of Q CT .Q/ implies CG.Pr / G or CG.Pr / Q CT .Q/ D D for r 1; 2; : : : ; s. If there is a Sylow pr -subgroup Pr such that Pr E G and D CG.Pr / Q CT .Q/, then, since CG.Pr / is normal in G, Q E G, a contradic- D tion. Thus Pr 6 Z.G/ if Pr E G. If Pr µ G, then NG.Pr / CG.Pr / and there- D fore G is pr -nilpotent. Let M be the intersection of all normal pr -complements of non-normal Sylow pr -subgroups Pr for r 1; 2; : : : ; s . Then M T N P 2 ¹ º \ D with N 6 Z.G/, and therefore P E G. So (4) is true. (5) By (1)–(3), G Q;CT .Q/; P . Lemma 2.3 implies that CT .Q/ is an D h i abelian group. Now we claim CP .Q/ 6 Z.G/. In fact, if ˆ.P / 1, then we D have CP .Q/ 6 Z.G/. If ˆ.P / 1, then, by (4), ˆ.P / 6 ˆ.G/ 6 Q CT .Q/ ¤ and therefore ˆ.P / 6 CP .Q/. It follows from Q CT .Q/ 6 CG.CP .Q// that CP .Q/ E G. If CG.CP .Q// Q CT .Q/, then it is easy to see Q E G, a con- D tradiction. Thus we still have CP .Q/ 6 Z.G/. If P is a non-abelian group, then, since p > 2, there exists a cyclic subgroup z h i of P such that z µ P . Thus G NG. z / is a power of p and NG. z / contains h i j W h i j h i a Hall p0-subgroup of G. It follows from CP .Q/ 6 Z.G/ that there is an element g g G such that .Q CT .Q// NG. z /. Since z is not contained in CT .Q/, 2 Ä h i NG. z / G, a contradiction. So (5) is true. h i D (6) It is clear that we may assume that t is an r-element with r a prime such that t E T but t µ G. In this case t CT .Q/. By (2) and (3), T CT .Q/ h i h i h i — j W j is a power of p and ŒP; Q 1. Since P is a normal abelian Sylow p-subgroup, ¤ Finite soluble groups 681

then, by [4, Theorem 5.2.3], P CP .Q/ ŒP; Q. It is clear that CT .Q/ nor- D malizes ŒP; Q and therefore ŒP; Q is a normal subgroup in G. The maximality of Q CT .Q/ implies that ŒP; Q is a minimal normal subgroup in G. The proof is now complete.

Theorem 3.5. Let q be the smallest prime dividing the order of a group G, and let Q and T be a Sylow q-subgroup and a Hall q0-subgroup in G respectively. Then G is a soluble non-nilpotent NCM-group if and only if G is one of the following type groups: (I) G Q T with Q a Dedekind group and T a soluble non-nilpotent D NCM-group. (II) G H .Q Ì P/ with H an abelian Hall p; q -subgroup, Q a Dede- D ¹ º0 kind group and P a Sylow p-subgroup. Furthermore, CP .Q/ is an abelian maximal subgroup in P and HP CQ.P / is maximal in G. If P is a non- abelian group, then there is an element c P such that P CP .Q/ c 2 D h i with P ac for any a CP .Q/. 0 Ä h i 2 (III) G H .K Ì Q/ with H an abelian Hall p; q -group and K a mini- D ¹ º0 mal normal subgroup of order pm in G, and there is a maximal subgroup Q1 of Q and an element b Q Q1 such that Q Q1 b with ab E Q 2 n D h i h i but ab µ G for any a Q1. Furthermore, every element of Q1 induces h i 2 a power automorphism on K and NG. x / NG. z / for any z K 1 h i D h i 2 ¹ º and for any x Q with x µ Q. 2 h i (IV) G H S .K ÌQ/ with H an abelian Hall p; q 0-group, S an abelian D ¹ mº p-group, K a minimal normal subgroup of order p in G, CQ.T / a Dede- kind group and CQ.T / maximal in Q. Furthermore, there exists an element b Q CQ.T / such that Q CQ.T / b with ab E Q but ab µ G 2 n D h i h i h i for any a CQ.T /. 2 (V) G H .P Ì Q/ with H an abelian Hall p; q -subgroup, P a cyclic D ¹ º0 Sylow subgroup of order p and Q an NCM-group. Furthermore, CQ.P / is normal in G and there exists an element c Q CQ.P / with the property 2 n that Q CQ.P / c and y E Q but y µ G for any y Q CQ.P /. D h i h i h i 2 n (VI) G .T Ì CT .Q// Ì Q with T a cyclic subgroup of order p, CT .Q/ an D 0 0 abelian Hall p; q -subgroup of G and Q a Dedekind group. ¹ º0 m (VII) G .T Ì CT .Q// Ì Q with T a minimal normal subgroup of order p D 0 0 in G, CT .Q/ an abelian Hall p; q -subgroup of G and Q a Dedekind ¹ º0 group. Furthermore, there is a maximal subgroup T1 of T and an r-element d T T1 with r a prime and r p such that T T1 d , every element 2 n ¤ D h i of Q and every element of T1 induces a power automorphism on T 0. 682 J. Cao and X. Guo

m (VIII) G .T Ì CT .Q// Ì Q with T a minimal normal subgroup of order p D 0 0 in G, CT .Q/ an abelian Hall p; q -subgroup of G and Q a Dedekind ¹ º0 group. Furthermore, there is a maximal subgroup Q1 of Q and an element e Q Q1 such that Q Q1 e , every element of Q1 and every element 2 n D h i of T induces a power automorphism on T 0. Proof. We ﬁrst assume that G is a soluble non-nilpotent NCM-group. In case that T 6 CG.Q/, then G Q T with T a soluble non-nilpotent group. By D Lemma 2.4, T is also an NCM-group. If there exists a subgroup x in Q such h i that x µ Q, then the maximality of NG. x / and Lemma 2.3 imply that T is h i h i an abelian group, and therefore G is a nilpotent group, a contradiction. Thus Q is a Dedekind group, and so G is the type (I). Now we assume T CG.Q/. By Lemma 3.2, we may consider the following — two cases. Case 1: Q E G. If every cyclic subgroup x in Q is normal in G, then the min- h i imality of q implies that T 6 CG.Q/. Thus there exists a subgroup y in Q such h i that y µ G. If Q is a Dedekind group, then, by Lemma 2.3, NG. y / Ty Q h i h i D with Ty an abelian Hall q -subgroup of NG. y /. Since T CG.Q/, we see 0 h i — CG.Q/ Ty Z.Q/ and Ty is normal in G, and therefore NG. y / Ty Q D h i D is normal in G. The maximality of NG. y / implies that Ty is maximal in T and h i Ty CT .Q/. Thus there is a prime p such that G NG. y / p and there D j W h i j D is a p-element c T such that T Ty c . If Q NG. c /, then ŒQ; c 1 2 D h i Ä h i D and therefore Q 6 CG.T /, a contradiction. Hence T NG. c /. It follows that Ä h i NG. c / TCQ.c/, and therefore the Hall p; q -subgroup H in G is con- h i D ¹ º0 tained in the center of G. Let G H .Q Ì P/ with P CP .Q/ c a Sylow D D h i p-subgroup in G. If P is a non-abelian group, then we have pn c > p2. D jh ij Noting that P ŒCP .Q/; c 6 CP .Q/ c 6 Z.P /, we see that there is an el- 0 D h i \h i ement g CP .Q/ such that P Œg; c . Now for any a CP .Q/, we see ac 2 0 D h i 2 h i is normal in P by the above proof. Thus Œg; c Œg; ac P ac and the group D 2 0 \h i is the type (II). Now we assume that Q is not a Dedekind group and that Q NG. y /. The — h i maximality of NG. y / implies that NG. y / contains a Hall q -subgroup K in G. h i h i 0 Without loss of generality, we may assume K T . By Lemma 2.3 and the nor- D mality of Q, we have NG. y / T Qy with T an abelian group and Qy a Sylow h i D q-subgroup of NG. y / and therefore Qy CQ.T /. Noting that h i D G CQ.T / CQ.T /; ŒCQ.T /; Q and ŒCQ.T /; Q 6 ˆ.Q/; D h i G G we see CQ.T / < Q. The maximality of CQ.T / T implies CQ.T / CQ.T /. D Let x be a cyclic subgroup of Q. If x is normal in G, then the minimal- h i h i ity of q implies that NG. x /=CG.x/ is a q-group and therefore x CQ.T /. h i 2 Finite soluble groups 683

g If Q NG. x /, then there is an element g G such that NG. x / T Qx. — h i 2 g h i D The maximality of NG. x / implies Qx CQ.T /. It follows that Qx CQ.T / h i D D and therefore x CQ.T /. Hence CQ.T / is a Dedekind group and x is not nor- 2 h i mal in G with Q NG. x / if x Q CQ.T /. Ä h i 2 n Let c Q CQ.T /. Then, by the above arguments and Lemma 2.3, we have 2 n NG. c / Tc Q with Tc a Hall q -subgroup in NG. c /. It follows from the h i D 0 h i commutativity of T that NG. c / E G and therefore Tc is maximal in T . Then h i Tc CT .Q/ and G NG. c / p is a prime. Thus QCT .Q/=CQ.T /CT .Q/ D j W h i j D q is a minimal normal subgroup in G=CQ.T /CT .Q/ and c CQ.T /. In case that 2 ŒCQ.T /; c 1, then, for any cyclic subgroup x of CQ.T /, we have D h i NG. x / T;CQ.T /; c G; h i D h i D in contradiction to that Q is not a Dedekind group. Thus ŒCQ.T /; c 1 and there- ¤ fore the order of c is at least q2. On the other hand, there is a p-element t in T such that T CT .Q/ t . Then D h i p 1 G t Y t j c c h i c : h i D h i D h i j 0 D G The maximality of T CQ.T / implies that Q CQ.T / c . t Dt h i Noting that c Q CQ.T /, we see that c is normal in Q. If there exists 2 n h i t k a positive integer k with 1 k q 1 such that c xc with x CQ.T /, Ä Ä D 2 then it is easy to see that t normalizes CQ.T / c and therefore CQ.T / c is h i h i h i normal in G. Hence Q CQ.T / c and QCT .Q/=CQ.T /CT .Q/ is a cyclic D h i group of order q. By the minimality of q, QCT .Q/=CQ.T /CT .Q/ is contained in the center of G=CQ.T /CT .Q/ and therefore T is normal in G, in contradiction to T CG.Q/. Thus we may assume that there is a positive integer l with — 1 l p 1 such that Ä Ä t t 2 t t l t t l 1 c CQ.T / c ; c CQ.T / c; c ; : : : ; c CQ.T / c; c ; : : : ; c … h i … h i … h i but t l 1 t t l c C CQ.T / c; c ; : : : ; c : 2 h i t t l For convenience, we set R c; c ; : : : ; c . Then it is easy to see that CQ.T /R is D h i q normal in G and therefore Q CQ.T /R. Furthermore we see CQ.T / R c D q \ D h i from the choice of l. In fact, if CQ.T / R c , then there is an element l 1 l q i\0 t ¤i1 h i t i t i r CQ.T / such that r c and r c .c / .c / l 1 .c / l . It follows 2 … h i D that there exists an i such that .i ; q/ 1. Without loss of generality, we may k k D assume k l. In this case D l l 1 l 1 t il t il 1 i0 t t .c / .c / c r CQ.T / c; c ; : : : ; c ; D 2 h i 684 J. Cao and X. Guo and therefore t l t t l 1 c CQ.T / c; c ; : : : ; c ; 2 h i t l t t l 1 in contradiction to c CQ.T / c; c ; : : : ; c . Noting that the order of every … h 2 i element in Q CQ.T / is at least q , we see that every element of order q in R n is contained in CQ.T / R and therefore there is a unique subgroup of order q \ in R. By Lemma 2.7, R is either a cyclic group or a generalized quaternion group. If R is a cyclic group, then T is normal in G by using the above proof, a contradiction. If R is a generalized quaternion group, then Z.R/ 2. Since m n j j D Œct ; ct 6 c2 6 Z.R/ for any 0 6 m; n 6 l, we see that R 6 c2 6 Z.R/ h i 0 h i and therefore R 2. So R is a quaternion group of order 8. Noting that CQ.T / j 0j D is a Dedekind 2-group, we may assume that CQ.T / Q8 L with Q8 a quater- D nion group and L an elementary abelian 2-group. If there exists a cyclic subgroup d in CQ.T / of order 4 such that d µ Q, then dc Q CQ.T / and h i 2 h i 2 n so dc E Q. Since c CQ.T / R 6 Z.Q/ and NG. d / T CQ.T /, h i h i D \ h i D we see c 1.dc/c cd dc , cd dc and d 1cd c 1. Thus .dc/2 d 2, D 2 h i ¤ D D d 1c .dc/3 cd and c2 d 2. It follows that D D D 2 Q ŒCQ.T /; CQ.T /; ŒCQ.T /; R; ŒR; R c 0 D h i D h i and therefore d E Q, a contradiction. Hence there is a cyclic subgroup u h i h i in CQ.T / of order 2 such that u µ Q. Since uc Q CQ.T /, we see that the h i 2 n order of uc is at least 4 by the above proof. On the other hand, uc cu implies ¤ that u 1cu c 1 and therefore uc is an element of order 2, a contradiction. These D contradictions tell us that Q must be a Dedekind group. Case 2: G is a q-nilpotent group. By Lemma 3.4 (1)–(3), there is a prime p such that T CT .Q/ is a power of p and Q CT .Q/ is maximal in G. Lemma 2.3 j W j implies that CT .Q/ is an abelian group. In order to complete the proof, we consider the following two subcases: (a) T is a nilpotent group. In this case, it is clear that CT .Q/ contains a Hall p0-subgroup H of T . The commutativity of CT .Q/ implies H 6 Z.G/. Let P be a Sylow p-subgroup of G. Then CT .Q/ 6 Z.G/ by Lemma 3.4 (5). By Lemma 3.4 (5) again, we have that every subgroup of P is normal in T . If there exists an element z P such that z µ G, then, by Lemma 3.4 (6), we 2 h i have G H CP .Q/ .K Ì Q/ with K a minimal normal subgroup in G and D K a p-group. Since NG. z / T Ì NQ. z / is maximal in G for any z K with h i D h i 2 z 1, we see that NQ. z / Q1 is maximal in Q and therefore NG. z / E G. ¤ h i D g h i It follows that NG. z / NG. z / for any g Q. Hence every element of Q1 h i D h i 2 induces a power automorphism on K. Let b Q Q1. Then we have Q Q1 b . 2 n D h i Lemma 3.4 (1) implies that ab E Q but ab µ G for any a Q1. If there is h i h i 2 Finite soluble groups 685

an element x Q with x µ Q, then, by Lemma 3.4 (1), K 6 CG.x/ and there- 2 h i fore NG. x / NG. z / by Lemma 2.3. If CP .Q/ 1, then G is a group of h i D h i D type (III). Now we assume that CP .Q/ 1. Let s CP .Q/ with order p, and sz be ¤ 2 h i a subgroup of order p for any z K. If g NQ. sz /, then there is a positive i g 2 g 2 h i i 1 integer i such that .sz/ .sz/ sz . It follows that s CP .Q/ K 1 D D 2 \ D and therefore g CQ.z/. By Lemma 2.6, we have CQ.z/ CQ.K/ 6 CQ.T /. 2 D Thus NQ. sz / 6 CQ.T /. If NQ. sz / Q, then Q 6 CG.T /, a contradiction. h i h i D So we may have NG. sz / T Ì NQ. sz / with NQ. sz / maximal in Q. Thus h i D h i h i NQ. sz / CQ.T / NQ. z / h i D D h i for any z K. It is easy to ﬁnd that there is an element b Q CQ.T / such that 2 2 n Q CQ.T / b with ab E Q but ab µ G. We have proved that D h i h i h i NG. x / NG. z / T CQ.T / h i D h i D for any x µ Q and any z K. Therefore CQ.T / is a Dedekind group. So G is h i 2 a group of type (IV). If z E G for any z P , then every element of Q induces a p -power au- h i 2 0 tomorphism on P . Applying Lemma 2.6, we see CP .Q/ P or CP .Q/ 1. D D It is clear that the former case implies that G is nilpotent. Thus CT .Q/ is a Hall p -subgroup of T . The maximality of Q CT .Q/ and the normality of every 0 subgroup of P in G imply that P is a cyclic subgroup of order p. Noting that g there is an element g G such that CG.P / T CQ.P / , we see that CQ.P / 2 D is normal in G. Since G=CG.P / is a subgroup of a cyclic group of order p 1, there is an element c in Q such that Q CQ.P / c . By Lemma 3.4 (1), we have D h i y E Q but y µ G for any y Q CQ.P /. It is clear that Q is an NCM-group h i h i 2 n by Lemma 2.4. So G is a group of type (V). (b) T is a non-nilpotent group Since T is not nilpotent, there is a prime r .T / with r p such that 2 ¤ a Sylow r-subgroup of CT .Q/ is not normal in T . It follows from the maximality of Q CT .Q/, normality of P and commutativity of CT .Q/ that CT .Q/ is just a system normalizer of T . It is clear that all Sylow subgroups of T are abelian. By Lemma 2.8, we have G .T Ì CT .Q// Ì Q with T a minimal normal subgroup. If there exists an D 0 0 element x Q with x µ Q, then NG. x / T Ì NQ. x /. By Lemma 2.3, 2 h i h i D h i T is an abelian group, a contradiction. So Q is a Dedekind group. If CP .Q/ 1, then sz is a subgroup of order p for any z T and s CP .Q/ ¤ h i 2 0 2 with order p. If g NG. sz /, then there is a positive integer i with the prop- i 2 g h gi i 1 erty that .sz/ .sz/ sz . It follows that s CP .Q/ T 0 1 and there- D D 2 \u D fore g CG.z/. If there exists an element u G such that Q 6 NG. sz /, then 2 2 h i 686 J. Cao and X. Guo

u we get z CT .Q /, a contradiction. Thus we have T 6 NG. sz / 6 CG.z/ for 2 h i any z T , and therefore T is a nilpotent group, a contradiction. So CT .Q/ is 2 0 a Hall p; q -subgroup of G. ¹ º0 If T p, then G is a group of type (VI). Now we assume that T > p. If j 0j D j 0j there exists an element z T such that z µ T , then there is a prime r .T / 2 0 h i 2 with r p such that G NG. z / is a power of r. Thus there is an element ¤ g j W h i j g G such that Q 6 NG. z /. Noting that T NG. z / and T T CT .Q/, 2 h i 0 Ä h i D 0 we see that NT . z / is normal in T and therefore NG. z / is normal in G. Thus h i h i NT . z / T1 is maximal in T and there is an r-element d T with the property h i D h 2 that T T1 d . Since NG. z / NG. z / for any h G, we see that every D h i h i D h i 2 element of T1 and every element of Q induce a power automorphism on T 0. So G is a group of type (VII). If there exists an element z T such that z E T , then 2 0 h i zg CT .Q/ zCT .Q/ g zg h i D h i D h i for any g Q. Thus every element of T induces a power automorphism on T . 2 0 Furthermore, the maximality of NG. z / implies that NQ. z / is maximal in Q h i h i and therefore NG. z / is normal in G. Thus h i g NQ. z / Q1 6 NG. z / NG. z / h i D h i D h i for any g Q and so every element of Q1 induces a power automorphism on T . 2 0 Let e be an element of Q with e Q Q1. Then Q Q1 e . So G is a group of 2 n D h i type (VIII). Conversely, it is clear that G is a soluble non-nilpotent group if G is one of the types from (I) to (VIII). Let n be any non-normal cyclic subgroup of G. h i If n is a q-group or a q -group and G is one of the types from (I) to (VIII), h i 0 then it is easy to see that NG. n / is maximal in G. So we may assume that h i n nq nq with nq 6 Q and nq 6 T . It is clear that at least one of h i D h i h 0 i h i h 0 i groups nq and nq is not normal in G. If one of the groups nq and nq h i h 0 i h i h 0 i is normal in G, then, without loss of generality, we may assume that nq µ G h i and nq E G. Since nq is characteristic in n , we see NG. n / 6 NG. nq / h 0 i h i h i h i h i and therefore NG. n / NG. nq / is maximal in G. Now assume that neither h i D h i nq nor nq is normal in G. Noting that NG. n / NG. nq / NG. nq / h i h 0 i h i D h i \ h 0 i and nq 6 CG.nq /, we see that NG. nq / NG. nq / is maximal in G if G h i 0 h i D h 0 i is one of the types from (I) to (VIII). Thus NG. n / NG. nq / NG. nq / is h i D h i D h 0 i maximal in G. The proof is now complete.

We immediately have the following corollary from Theorem 3.5.

Corollary 3.6. Soluble NCM-groups have Fitting length at most 2. Finite soluble groups 687

Acknowledgments. The authors would like to thank the referee for his/her valu- able suggestions and useful comments which contributed to the ﬁnal version of this paper.

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Received May 14, 2013; revised November 27, 2013.

Author information Jianji Cao, Department of Mathematics, Shanghai University, Shanghai 200444, P.R. China; and Department of Mathematics, Shanxi Datong University, Datong Shanxi 037009, P.R. China. Xiuyun Guo, Department of Mathematics, Shanghai University, Shanghai 200444, P.R. China. E-mail: [email protected]