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 re- search 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 nor- malizer 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 struc- ture 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 sub- group 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-sub- group. 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 sub- groups 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.