On a Class of Finite Soluble Groups

J. Group Theory 21 (2018), 839–846 DOI 10.1515/jgth-2018-0015 © de Gruyter 2018

On a class of ﬁnite soluble groups

Adolfo Ballester-Bolinches, John Cossey and Yangming Li Communicated by Alexander Olshanskii

Abstract. The aim of this paper is to study the class of finite groups in which every subgroup is self-normalising in its subnormal closure. It is proved that this class is a subgroup-closed formation of finite soluble groups which is not closed under taking Frattini extensions and whose members can be characterised by means of their Carter subgroups. This leads to new characterisations of finite soluble T-, PT- and PST-groups. Finite groups whose p-subgroups, p a prime, are self-normalising in their subnormal closure are also characterised.

1 Introduction

All groups considered in this paper will be ﬁnite. It has long been known that permutable and subnormal subgroups play an important role in the structural study of the groups, and in recent years there has been a considerable interest in the phenomenon of subgroup permutability. Recall that a subgroup H is said to be permutable in a group G if HK KH for all D subgroups K of G, and Sylow permutable or S-permutable if HP PH for all D Sylow subgroups P of G. According to a known result of Kegel, S-permutable subgroups are always subnormal ([1, Theorem 1.2.14]), and the intersection of S-permutable (respectively, subnormal) subgroups of a group G is again S-permutable (respectively, subnormal) in G (see [1, Theorem 1.2.19] and [3, Corol- lary A.14.2]). Therefore, the intersection BG.H / (respectively, SG.H /) of all S-permutable (respectively, subnormal) subgroups of a group G containing a subgroup H is the smallest S-permutable (respectively, subnormal) subgroup of G containing H which is called the S-permutable closure (respectively, subnormal) closure of G.

The ﬁrst author is supported by the grant MTM2014-54707-C3-1-P from the Ministerio de Economía y Competitividad, Spain, and FEDER, European Union and Prometeo/2017/057 of Gen- eralitat (Valencian Community, Spain). The third author has been supported by the project of NSF of China (11271085) and NSF of Guangdong Province (China) (2015A030313791) and Ma- jor Projects in Basic Reaearch and Applied Research (Natural Science) of Guangdong Province (2017KZDXM058).

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Unfortunately, the permutable closure AG.H / of H in G, that is, the intersection of all permutable subgroups of G containing H is not permutable in general ([2, Example 3]). In [2, Theorem 4], it is proved that if H is self-normalising in AG.H /, then AG.H / is a permutable subgroup of G and G is a soluble PT-group, that is, every subnormal subgroup of G is permutable, if and only if every subgroup is self-normalising in its permutable closure [2, Theorem 8]. A characterisation theorem for soluble PST-groups, or groups in which every subnormal subgroup is S-permutable, was showed in [2, Theorem 4]: G is a soluble PST-group if and only if every subgroup of G is self-normalising in its S-permutable closure. Both results follow the pattern established in [4] for soluble T-groups, or groups in which every subnormal subgroup is normal. These groups are characterised as those groups in which every subgroup is self-normalising in its normal closure. The results just mentioned suggest that the investigation of groups in which every subgroup is self-normalising in its subnormal closure is a natural next objec- tive. This is the main aim of the present paper.

Deﬁnition 1. (1) A subgroup H of a group G is said to be c-embedded in G if H is self-normalising in SG.H /. (2) A group G is said to be a C-group if every subgroup is c-embedded in G.

Our ﬁrst result conﬁrms that the class of all C-groups is a formation of soluble groups.

Theorem A. The class of all C-groups is subgroup-closed formation of soluble groups of p-length at most 1 for all primes p.

Note that not every C-group is supersoluble as the alternating group A4 of degree 4 shows. Moreover, it is not closed under taking Frattini extensions in general. It is enough to consider the group G SL.2; 3/ which is not a C-group but D G=ˆ.G/ is a C-group. It must be noted that the structure of the C-groups does not resemble that of soluble T-, PT- or PST-groups (see [1, Chapter 2]). For instance, there are C-groups G whose nilpotent residual is neither nilpotent nor a Hall subgroup of G.

Example 2. (1) Let A be a cyclic group of order 4 and let B be a cyclic group of order 3 on which A acts invertingly. The extension BA has a faithful irreducible module N of dimension 2 over the ﬁeld of 7 elements. Then the semidirect product G NBA is a C-group whose nilpotent residual is not nilpotent. D (2) Let G D26 A4, where D26 is a dihedral group of order 26 and A4 is an D alternating group of order 4. Then G is a C-group whose nilpotent residual is not a Hall subgroup of G.

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Recall that a Carter subgroup of a group is a nilpotent self-normalising subgroup of G. A classical result by Carter assures that every soluble group has a unique conjugacy class of Carter subgroups [3, Theorem III.4.6]. Our next theorem shows that the class of all C-groups can be characterised by means of their Carter subgroups.

Theorem B. A group G is a C-group if and only if G is soluble and every Carter subgroup of every subgroup H of G is also a Carter subgroup of SG.H /. This theorem has two interesting consequences. The ﬁrst one shows that it is enough to consider nilpotent subgroups to prove that a group belongs to the class of all C-groups.

Corollary 3. A group G is a C-group if and only if every nilpotent subgroup of G is self-normalising in its subnormal closure. Our second corollary shows that the classes of all soluble T-, PT- and PST- groups can be characterised by means of their Carter subgroups.

Corollary 4. The following statements hold. (1) A soluble group G is a T-group if and only if every Carter subgroup of every subgroup H of G is also a Carter subgroup of H G, the normal closure of H in G. (2) A soluble group G is a PT-group if and only if every Carter subgroup of every subgroup H of G is also a Carter subgroup of AG.H /. (3) A soluble group G is a PST-group if and only if every Carter subgroup of every subgroup H of G is also a Carter subgroup of BG.H /.

2 Proofs

Our ﬁrst results analyse some closure properties of the class of all C-groups. We begin with the following elementary observation. Let A be a subgroup of the group G and suppose that A is self-normalising in its subnormal closure X. Let B a subgroup of G containing A. By [3, Lemma A.14.1], B X contains SB .A/. \ Since NB X .A/ A, it follows that NSB .A/.A/ A. \ D D Lemma 5. The class of all C-groups is a subgroup-closed class of groups which is closed under taking epimorphic images.

Proof. By the above observation, the class of all C-groups is subgroup-closed. We prove now that if G is a C-group and N is a normal subgroup of G, then G=N is

Brought to you by | The Australian National University Authenticated Download Date | 1/30/19 1:59 AM 842 A. Ballester-Bolinches, J. Cossey and Y. Li also a C-group. Let H=N be a subgroup of G=N . Then H is c-embedded in G. Therefore, if X SG.H /, then NX .H / H. By [3, Lemma A.14.1], X=N is the D D subnormal closure of H=N in G=N . Moreover, N .H=N / H=N . Therefore X=N D H=N is c-embedded in G=N and hence G=N is a C-group. Groups in which every p-subgroup, p a prime, is c-embedded have a restricted structure. We call them Cp-groups.

Lemma 6. Let p be a prime and let G be a Cp-group.

(1) If A is a subgroup of G, then A is a Cp-group.

(2) If L is a normal p-subgroup of G, then G=L is a Cp-group.

(3) If L is a normal p0-subgroup of G, then G=L is a Cp-group. Proof. Statement (1) follows by the observation at the beginning of the section and statement (2) can be proved by using the same arguments to those used in Lemma 5. (3) Arguing by induction on G , we may assume that L is a minimal normal j j subgroup of G. Let X=L be a p-subgroup of G=L. Then X YL for some Sylow D p-subgroup Y of X. Since G is a Cp-group, it follows that NZ.Y / Y , where D Z SG.Y /. Let T=L S .X=L/. Then T ZL. Assume that T is a proper D D G=L D subgroup of G. By statement (1), T is a Cp-group. Hence T=L is a Cp-group by induction. Since T=L S .X=L/, it follows that N .X=L/ X=L. D T=L T=L D Assume now that G T ZL. Then ŒZ; L is normal in G. It follows from D D [3, Lemma A.14.3] that L normalises Z. Since L is minimal normal in G, we have either G Z or L CG.Z/. D Â g Let gL NG=L.X=L/. We may assume that g Z. Then X Y L and there 2 gl 2 D exists l L such that Y Y . Thus gl NG.Y /. If G Z, then NG.Y / Y 2 D 2 D D and so g X. If L CG.Z/, it follows that g NZ.Y / Y . Hence, in both 2 Â 2 D cases, we have that gL X=L. 2 Therefore G=L is a Cp-group.

Theorem 7. Let p be a prime. If G is a Cp-group, then the p-length of G is at most 1.

Proof. Assume that G is a Cp-group. We prove that the p-length of G is at most 1 by induction on the order of G. Suppose that the largest p0-normal subgroup L Op .G/ is non-trivial. By Lemma 6, G=L is a Cp-group. Therefore G=L has D 0 p-length at most 1 by induction. Thus G has p-length at most 1. Consequently, we may assume that Op .G/ 1. 0 D Let A and B be maximal normal subgroups of G. By Lemma 6, A and B are both Cp-groups. The induction hypothesis implies that A and B are of p-length

Brought to you by | The Australian National University Authenticated Download Date | 1/30/19 1:59 AM On a class of ﬁnite soluble groups 843 at most 1. Suppose that A B. Then G AB. Since the class of all groups ¤ D of p-length at most 1 is closed under taking normal products, it follows that G has p-length at most 1. Consequently, we may assume that G has a unique maximal normal subgroup, M say, and the p-length of M is at most 1. Since Op .M / Op .G/ 1, it follows that M has a normal Sylow p-subgroup. Let P 0 Â 0 D be a Sylow p-subgroup of G. If P were contained in M , then P would be normal in G and so G would have p-length at most 1. Therefore we may assume that P is not contained in M . Since M is the unique maximal normal subgroup of G, it follows that SG.P / G and so NG.P / P . D D Suppose that PM is a proper subgroup of G. By Lemma 6, PM is a Cp-group. By induction, PM has p-length at most 1. Then Op .PM / Op .M / 1 and 0 Â 0 D so P is a normal subgroup of PM . But P is self-normalising in PM . Hence M is contained in P . Assume that a minimal normal subgroup of G is not contained in M . Then G is a simple group. If G were non-abelian, then the subnormal closure of every non-trivial subgroup of G would be equal to G and so either G is a p0-group (and so G is of p-length at most 1), or every non-trivial p-subgroup of G would be self-normalising. In the latter case, every Sylow p-subgroup of G would be cyclic of prime number. This cannot be happen. Hence G is abelian and so it is cyclic of order p. Therefore we may assume that Soc.G/ is contained in M . Therefore every minimal normal subgroup of G is an elementary abelian p-subgroup of G. Let N be a minimal normal subgroup of G. By Lemma 6, we have that G=N satisﬁes the hypothesis of the theorem. Hence G=N has p-length at most 1. Since the class of all groups of p-length at most 1 is a saturated formation by [3, Example A. 3.4 (b)], we may assume that G is has unique minimal normal subgroup, N say, and there exists a core-free maximal subgroup X of G such that G NX, N X 1 and D \ D N F.G/ (see [3, Theorem A.15.2]). Consequently, M N and X is a simple D D group of p-length at most 1. This implies that either X is a p-group or a p0-group. In particular, G has p-length at most 1. Assume that G PM . Then G=M is a cyclic group of order p. Since P M D \ is normal in G, it follows that G has p-length at most 1. The proof of the theorem is now complete.

Theorem 8. Let p be a prime and let G be a group. Then G is a Cp-group if and only if the following two conditions hold: (1) G is p-soluble of p-length at most 1,

(2) for any p-subgroup H of G, we have that C p .H / 1. O .SG .H // D Proof. Let H be a p-subgroup of G. Set X SG.H /. Since H is contained in D Op0 .X/ and Op0 .X/ is subnormal in G, it follows that X Op0 .X/. If X has D

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p p-length at most 1, it follows that O .X/ is a normal Hall p0-subgroup of X. Note that H is subnormal in a Sylow p-subgroup of X. Therefore Op.X/H is subnormal in X. Thus Op.X/H is subnormal in G and so X Op.X/H . D Assume that G is a Cp-group. By Theorem 7, G is p-soluble of p-length at p most 1. Then O .X/ is a normal Hall p -subgroup of X. Since NX .H / H, it 0 D follows that C p .H / 1. O .X/ D Assume now that G satisﬁes statements (1) and (2), and let H be a p-subgroup p of G. Then X O .X/H. Therefore we have that NX .H / H N p .H / D D O .X/ D H C p .H / H. Hence G is a Cp-group. O .X/ D Proof of Theorem A. By Lemma 5, the class of all C-groups is closed under taking epimorphic images. We shall prove next that it is closed under taking direct products. Assume this is not true and derive a contradiction. Then there exist C-groups G and H such that E G H is not a C-group. Let us take E of minimal order. D Then E has a subgroup S such that NX .S/ S, where X SE .S/. ¤ D If either S G or S H, then X SG.S/ or X SH .S/. In both cases, S is Â Â D D self-normalising in X, against the choice of E. Then neither G nor H contains S. Let U and V be the projections of S onto G and H, respectively. Then we have U 1 V and SE .S/ is contained in SG.U / SH .V / by [3, Theorem A.14.4]. 6D 6D If .x; y/ N .S/ S with x SG.U /; y SH .V /, then we have that 2 SE .S/ n 2 2 .s; t/.x;y/ .sx; t y/ S and so sx U for all s U and t y V for all t V . D 2 2 2 2 2 Since G is a C-group, we have x U . Analogously, we obtain y V and so 2 2 N .S/ U V . Moreover, there exists x U such that .x ; y/ S. There- SE .S/ Â 0 2 0 2 fore .x x 1; 1/ N .S/ S. Write z x x 1. Then z U . 0 2 SE .S/ n D 0 2 Set A SG.U / and B SH .V /. Since S.A B/.S/ is subnormal in E, it fol- D D lows that S.A B/.S/ SE .S/. The minimality of E gives E A B SE .S/ D D D and A G and B H. Let C and D be the nilpotent residuals of A and B, D D respectively. Then C D is the nilpotent residual of E by [3, Lemma II.2.12]. Since .C D/S is a subnormal subgroup of E containing S, it follows that E SE .S/ .C D/S. D D Clearly E is not nilpotent. Thus we can assume without loss of generality that C 1. Let M be a minimal normal subgroup of G contained in C . Since G is ¤ soluble by Theorem 8, it follows that M is an elementary abelian p-group for some prime p. By Lemma 5, G=M is a C-group. If M M 1 E, then E=M D Â Š .G=M / H is a C-group. Since .z; 1/M =M NE=M .SM =M /, it follows 2 that .z; 1/M SM =M . Hence .z; 1/ SM S so that .z; 1/ .m; 1/s, for 2 2 n D some elements m M and s S. 2 2 Suppose that F.G/ is not a p-group, and let N be a minimal normal q-subgroup of G for some prime q p. Arguing as above, we conclude that .z; 1/ .n; 1/t, ¤ D for some elements n N and t S. Hence we have .mn 1; 1/ ts 1 S, giv- 2 2 D 2

Brought to you by | The Australian National University Authenticated Download Date | 1/30/19 1:59 AM On a class of finite soluble groups 845 ing 1 .mq; 1/ ..mn 1/q; 1/ S and then .z; 1/ .m; 1/s S, contrary to 6D D 2 D 2 the choice of z. Therefore F.G/ is a p-group. Since G has p-length at most 1 and Op .G/ 1, we have that F.G/ is a Sylow 0 D p-subgroup of G centralising M . Thus E=CE .M / G= CG.M / is a p -group. Š 0 In particular, S=CS .M / is a p0-group so that M is a semisimple module by [3, Theorem A.11.4]. Then M .M S/ T for some subgroup T T 1 D \ D normalised by S. Clearly we may assume that .z; 1/ T . Therefore we have 2 Œ.z; 1/; S T S 1. In particular, z CG.U /. Assume that U is a p-group. Ä \ D 2 Then U is contained in F.G/ and so it is subnormal in G. Since A G, it follows D that G U F.G/, contrary to the choice of G. Consequently, U has a non- D D trivial Hall p -subgroup, X say. Then U .U F.G//X. Let D SG.X/. Then 0 D \ D F.G/D is a subnormal subgroup of G containing U . Therefore G F.G/D. D Applying [3, Lemma II.2.12], C is the nilpotent residual of D and, since CX is subnormal in G, it follows that D CX. Thus z ND.X/ X and X is not D 2 n c-embedded in G. This final contradiction confirms our claim. Assume now that G is a group with two normal subgroups E and F such that G=E and G=F are C-groups and E F 1. Then G is isomorphic to a subgroup \ D of the direct product G=E G=F which is a C-group by the above argument. Applying Lemma 5, we have that G is a C-group. Assume now that G is a C-group. Then G has p-length at most 1 for all primes p. Hence G is p-soluble for all primes p so that G is soluble.

Proof of Theorem B. Assume that G is a C-group. By Theorem A, G is soluble. Let C be a Carter subgroup of a subgroup H of G. Then C is self-normalising in H and so if L is the nilpotent residual of H, we have H LC . Let X SG.H / D D and let N be the nilpotent residual of X. Then X contains Y SG.C /. Note that D H is not contained in any proper normal subgroup of X so that X HN . Hence D X NC and hence C is not contained in any proper normal subgroup of X. In D particular, X Y . Since NY .C / C , it follows that C is a Carter subgroup of X. D D Assume now that G is a soluble C-group such that every Carter subgroup of every subgroup is also a Carter subgroup of its subnormal closure. Let H be a subgroup of G. Then H contains a Carter subgroup C of X SG.H /. Let g D g NX .H /. Then C and C are Carter subgroups of H. By [3, Theorem III.4.6], 2 gh there exists h H such that C C . Hence gh NX .C / C . In particular, 2 D 2 D g H. Therefore NX .H / H and G is a C-group. 2 D Proof of Corollary 3. Only the sufﬁciency of the condition is in doubt. Assume that every nilpotent subgroup of a group G is self-normalising in its subnormal closure. Applying Theorem 8, G is soluble. Let H be a subgroup of G. Arguing as in the proof of Theorem B, we have that X SG.H / SG.C / for all Carter D D

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subgroups C of H. Since C is nilpotent, it follows that NX .C / C . Hence C is D also a Carter subgroups of X. By Theorem B, G is a C-group.

Proof of Corollary 4. We give a proof for the third statement. If G is a soluble PST-group, then SG.H / is S-permutable for all subgroups H of G and so we have SG.H / BG.H /. By Theorem B, every Carter subgroup of H is also a Carter D subgroup of BG.H /. Conversely, let H be a subgroup of G and let C a Carter subgroup of H. Then z C is a Carter subgroup of Z BG.H /. If z NZ.H /, then C and C are Carter D 2 subgroups of H. By [3, Theorem III.4.6], there exists h H such that C zh C . 2 D Hence zh NX .C / C . In particular, z H . Applying [2, Theorem 13], we 2 D 2 conclude that G is a PST-group. Note that if G is a soluble T-group, then every subnormal subgroup is normal. Hence SG.H / BG.H /. If G is a soluble PT-group, then every subnormal D subgroup is permutable and SG.H / AG.H / by [2, Theorems 4 and 8]. Hence D statements (1) and (2) follow using similar arguments.

Bibliography

[1] A. Ballester-Bolinches, R. Esteban-Romero and M. Asaad, Products of Finite Groups, De Gruyter Exp. Math. 53, Walter de Gruyter, Berlin, 2010. [2] A. Ballester-Bolinches, R. Esteban-Romero and Y. Li, On self-normalizing subgroups of ﬁnite groups, J. Group Theory 13 (2010), 143–149. [3] K. Doerk and T. Hawkes, Finite Soluble Groups, De Gruyter Exp. Math. 4, Walter de Gruyter, Berlin, 1992. [4] Y. Li, Finite groups with NE-subgroups, J. Group Theory 9 (2006), 49–58.

Received February 13, 2018.

Author information Adolfo Ballester-Bolinches, Department of Mathematics, Guangdong University of Education, 510310, Guangzhou, P.R. China; and Departament de Matemàtiques, Universitat de València, 46100 Burjassot, València, Spain. E-mail: [email protected] John Cossey, Australian National University, Canberra, ACT 2601, Australia. E-mail: [email protected] Yangming Li, Department of Mathematics, Guangdong University of Education, 510310, Guangzhou, P.R. China. E-mail: [email protected]

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