Subgroups Permutable with All Sylow Subgroups

JOURNAL OF ALGEBRA 207, 285᎐293Ž. 1998 ARTICLE NO. JA987429

Peter Schmid

Mathematisches Institut, Uni¨ersitat¨¨ Tubingen, Auf der Morgenstelle 10, D-72076, Tubingen,¨ Germany

View metadata, citation and similar papersCommunicated at core.ac.uk by Walter Feit brought to you by CORE provided by Elsevier - Publisher Connector Received November 4, 1997

1. INTRODUCTION

Fix a finite group G. Whenever ⌺ is a nonempty set of subgroups of G let ⌺H denote the set of all subgroups H of G permutable with all S g ⌺ Ž.so that the complex product HS s SH is their join . It is obvious that ⌺H is closed with respect to joins. In case ⌺ s SylŽ.G is the set of all Sylow subgroups of G Ž.for all primes , it is also closed with respect to intersec- tionsŽ Kegelwx 4.Ž so that Syl ŽG .H is a lattice. . The present paper may be seen as continuation of Kegel’s work; to be more precise, it aims to continue the first part of his work, the second one being completed by Kleidmanwx 5 . SylŽ.G H is the full subgroup lattice of G precisely when G g ᑨ is nilpotent. The formation ᑨ of nilpotent groups will play a central role in general. As usual we denote by Gᑨ the nilpotent residual of x K G. For subgroups H and K of G we write HKxs F g K H and H s x x ² Hx< g K :Žfor the K-core and the K-closure of H, respectively H s xHxy1 .. H K H Suppose H g SylŽ.G . Then HK and H do belong to SylŽ.G , for all K, but H ᑨ usually does notŽ nor does it the commutator group HЈ nor the centre ZHŽ... On the other hand, we know that there is a unique subgroup of H minimal subject to containing H ᑨ and belonging to SylŽ.G H . This is always a normal subgroup of G:

THEOREM A. If H g SylŽ.GH is a subgroup of G permutable with all ᑨ G G ᑨ Sylow subgroups of G, thenŽ H . s Ž H. is the smallest subgroup of H containing Hᑨ and belonging to SylŽ.G H .

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0021-8693r98 $25.00 Copyright ᮊ 1998 by Academic Press All rights of reproduction in any form reserved. 286 PETER SCHMID

ᑨ ᑨ In particular, H g SylŽ.G H if and only if H is normal in G.Itis immediate from Theorem A that H is a subnormal subgroup of G.We ᑨ even can show that the normalizer NHGŽ.= G so that the chain of iterated normalizers reaches G Ž.see Proposition C . This also gives that H NHGŽ.g Syl Ž.G . Recall that, for any subnormal subgroup H of G,bya result of Wielandtwx 8 , H ᑨ is permutable with all subnormal subgroups of G. So in Theorem A the join Ž H ᑨ .G is indeed the product of the G-conjugates of H ᑨ. Usually SylŽ.G H is a proper sublattice of the lattice of subnormal subgroups of G. In order to investigate this more closely one may ask for conditions forcing normality. Following Rosebladewx 7 let us call two groups H and K orthogonal provided there is no nontrivial pairing into a third group Ž.HrHЈ m KrKЈ s 0 . Orthogonality is a necessary and suffi- cient condition for H and K in order to permute whenever they are embedded as subnormal subgroups in some common group. In our situation the implication is much stronger:

THEOREM B. Let H g SylŽ.G H . Then H is normalized by any subgroup of G orthogonal to HrHG. In particular, H is normal in G if the commutator factor groups GrGЈ and HrHЈ ha¨e relati¨ely prime orders.

Recall that a subgroup H of G is in the hypercentre ZGϱŽ.Žs last member of the ascending central series of G. precisely when H normalizes each Sylow subgroup of G. In case H is a permutable Ž.quasinormal subgroup of G, i.e., permutable with all subgroups of G, one knows that

HrHGGis in the hypercentre of GrH wx6 . This is not valid for arbitrary H g SylŽ.G H . On the other hand, this holds true if one assumes in addition permutability with all Sylow normalizersŽ seewx 1 and Proposition D below. . More spectacular is the following result:

THEOREM C. Assume that G is a sol¨able group and that H g SylŽ.G H . Then HrHG : ZGϱŽ.rHG if and only if H permutes with some system normalizer of G.

System normalizers Ž.ᑨ-normalizers have been introduced and studied by P. Hall. They form a distinguished conjugacy class of nilpotent subgroups which cover each central chief factor of the underlying solvable group and avoid the eccentric ones. Permutability with all system normalizers is a necessary condition for a subgroup in order to be hypercentrally embedded. PERMUTABLE SUBGROUPS 287

2. THE BASIC LEMMAS

Throughout G denotes a finite group. The reader is referred towx 2, 3 for the necessary background. For convenience we summarize some basic statements, mostly used without further comment: Ž.a If the subgroup H of G permutes with subgroups X and Y then it also permutes with their join ²:X, Y . Ž.b If the subgroups H and K permute then, for any prime p, some Sylow p-subgroup of H permutes with some Sylow p-subgroup of K w3, VI.4.7x . Ž.c Suppose N is a normal subgroup of G.If H g SylŽ.G H then HNrN g SylŽ.GrN H , and the converse holds in case N : H.IfŽ S is a Sylow subgroup of G, then SNrN is one of GrN, and if N : H then SH s Ž.SN H is a subgroup of G if and only if SH s HS. . Ž.d Syl ŽG .H is preserved underŽ. inner automorphisms of G.If H H g SylŽ.G and G0 is a subgroup of G containing H, then H g H SylŽ.G0 Ž.Sylow’s theorem and the Dedekind modular law . Ž.e Let H be a p-subgroup of G for some prime p. Then H is subnormal in G if and only if H : OGpŽ.. Ž.f Let H be a subgroup of G with p-power index, p a prime. Then p H is subnormal in G if and only if H = OGŽ.. p As usual OGpŽ.is the largest normal p-subgroup of G and OGŽ.the smallest normal subgroup of G with p-factor group. Note that OGpŽ.is the join of all pЈ-elementsŽ. Sylow q-subgroups for q / p of G. LEMMA A. Let H be a p-subgroup of G for some prime p. Then H g H p SylŽ.G if and only if NGŽ. H = OGŽ.. Proof. Let H g SylŽ.G H . Then HP s PH s P for each Sylow p-subgroup P of G. Hence H : OGpŽ.is subnormal in G. Moreover, if Q g Syl qŽ.G for some prime q / p, then H is subnormal, even normal, in p HQ s QH. Thus NHGŽ.= OGŽ.contains all pЈ-elements of G. p Conversely, if NHGpŽ.= OGŽ.then H : OGŽ.is subnormal in G and normalized by each Sylow q-subgroup of G, q / p.

LEMMA B. Let again H be a p-subgroup of G, p a prime. Then H normalizes each Sylow subgroup of G if and only if the centralizer CGŽ. H = OGpŽ.. Proof. If H normalizes each Sylow subgroup of G, by Lemma A it centralizes all Sylow q-subgroups of G for q / p. Conversely, if CHGŽ.= p OGŽ.then H : OGpŽ.is subnormal in G and normalizes each Sylow subgroup of G. 288 PETER SCHMID

p G Observe that in Lemma B we have CHGŽ.= OGŽ.precisely when H is centralized by OGpŽ.. Thus G induces a p-group of automorphisms on the p-group H G and so stabilizes a chain of subgroups. In other words, we have H : ZGϱŽ.. In fact, the following Ž certainly known . equality holds true: ZG NS. ϱŽ.s F G Ž. SgSylŽ.G

3. THE BASIC PROPOSITIONS

We wish to characterize the subgroups in SylŽ.G H by conditions on their structure and their embedding in G Ž.modulo G-core . This will be prepared in this section and carried out in the next one.

H G PROPOSITION A. If H g SylŽ.G then H rHG is nilpotent. Proof. Assume Ž.G, H is a counterexample with <characteristic subgroup of H is in Syl Ž.G H .

Proof. Let Hp be the Sylow p-subgroup of the nilpotent group H,for some prime p. Of course, Hp is a characteristic subgroup of H. Let further S g Syl qŽ.G for some prime q. PERMUTABLE SUBGROUPS 289

Ž.i « Žii . . By Proposition A we know that H and, therefore, Hp is subnormal in G. Thus Hp : S in case p s q. Moreover, in the case p / q, Hp is a subnormal Hall subgroup of HS s SH. Hence Hp is even normalized by S in the latter case. Ž.ii « Žiii . . From hypothesis Ž. ii and Lemma A we infer that NXG Ž .= p OGŽ.for each characteristic subgroup X of Hp. Lemma A once again shows that X g SylŽ.G H . Ž.iii « Ž.i . This is trivial.

4. SOME CONSEQUENCES

The preceding lemmas and propositions are crucial for our study of SylŽ.G H . We now derive some consequences and, in particular, establish Theorem BŽ. stated in the Introduction .

PROPOSITION C. The subgroup H of G belongs to SylŽ.GH if and only if for each prime p there is a Sylow p-subgroup HpGrHofHrH G such that p NHGpŽ.= OGŽ..

Proof. By Proposition A we know that HrHG is nilpotent. Hence the result follows by combining Proposition B and Lemma A. COROLLARY 1Ž Kegelwx 4.Ž. . Syl GH is a sublattice of the lattice of subnormal subgroups of G. Proof. By Proposition A, SylŽ.G H consists of subnormal subgroups of G. By virtue of StatementŽ.Ž a in Section 2 . we only need show that H H SylŽ.G is closed with respect to intersections. Let Hi g SylŽ.G for i s 1, 2, and let Pi be the inverse image in G of the Sylow p-subgroup of HiiGrŽ.H for some prime p. We may assume that Ž.Ž.H12l H G s H1 G l Ž.H2 G s 1. Then P12l P is a p-group. It must be the Sylow p-subgroup p of H12l H , by subnormality. Now use that OGŽ.normalizes the Pi and hence their intersection.

H H COROLLARY 2. H g SylŽ.G « NHGŽ.g Syl Ž.G .

Proof. Without loss we may assume that HG s 1. Also, in view of Corollary 1, it suffices to study the situation where H is a p-group for p some prime p. But then NHGŽ.= OGŽ.contains each Sylow q-subgroup of G for q / p, and G s NHPGŽ.for each Sylow p-subgroup P. We mention that in Corollary 2 the converse does not hold, in general, even if one assumes that H is cyclic. COROLLARY 3. Let H g SylŽ.G H , and let ␲ be the set of primes di¨iding ᑨ ␲ <

Proof. Use that HrHGGis nilpotentŽ.Ž. Proposition A and that NHis the intersection of the normalizers of the inverse images in G of the Sylow ᑨ ␲ pŽ. subgroups of HrHGp. Of course, G s F g␲ OG. ᑨ ᑨ It is obvious that K ␲ : G ␲ for any subgroup K of G. Also, H and K ᑨ ᑨ are orthogonal if and only if < HrH <

5. PROOF OF THEOREM A

ᑨ x x ᑨ Note that Ž H . s Ž H . for all x g G. It therefore follows from Proposition A and Satz 1.5 in Wielandtwx 8 that

ᑨ G G ᑨ Ž.H s Ž.H : H. We proceed by way of contradiction, assuming that there is a proper ᑨ G ᑨ subgroup X of Ž H . such that X = H and X g SylŽ.G H . Replacing X by XH , if necessary, we may assume that X is normal in H. There is a prime p such that OHpŽŽ ᑨ .G . Xis a proper subgroup of ᑨ G ᑨ G Ž H . ŽŽbelonging to Syl G.H.Ž. Thus we may assume also that H . rX ᑨ G is a p-group. Note that then Ž H . rXG is a p-group as well, because p-groups are residually closed. On the other hand, HrXG is not nilpotent. Without loss we may assume that XG s 1. We shall produce a contradiction by showing that H is yet nilpotent. Since X is a p-group belonging H p to SylŽ.G , by Lemma A we have NXGŽ.= OGŽ.. Let P be a Sylow p p-subgroup of G. Then X : OGpŽ.: P. Moreover, from G s OGPŽ. we infer that XPGs X s 1. p Let Y s OHXŽ.. Then YrX is the p-complement of the nilpotent group HrX. In particular, Y l P s X. Note that H is subnormal in p p HP s PH. Thus OHŽ.s OHPŽ.is normal in HP and, therefore, p OHŽ.l P normal in P. On the other hand,

p O Ž.H l P : Y l P s X. p We conclude that OHŽ.l P : XP s 1. However, since HrX is nilpotent and X is a p-group, this implies that H is nilpotent. This is the desired contradiction, completing the proof of Theorem A.

22n x 1 2 ny 1 EXAMPLE. Let P s ² x, yx< s y s 1, y s y q : for some integer n 1 2 n G 3Ž so that <

Since x Ž.of order 2 is not central in P and V is faithful, V² x: s U [ U0 where U0 is trivial and U is the unique nontrivial irreducible ކq²:x -module ᑨ Ž.up to isomorphism . We infer that U s HЈ s H and that U0 s ZHŽ.s ZHϱŽ.. P P Since V s U s U00, both U and U do not permute with the Sylow 2-subgroup P of G. Note also that NUGGŽ.s NU Ž0 .is a normal subgroup of G with index 2.

6. SYLOW NORMALIZERS

Recentlywx 1 it has been shown that a core-free subgroup of G is in the hypercentre of G provided it permutes with all Sylow subgroups and all Sylow normalizers of G. This is explained by the following.

H PROPOSITION D. Let H g SylŽ.G with HGGs 1. Let N s NŽ. S for some Sylow subgroup S of G. If HN s NH then H : N.

Proof. We know that H is nilpotentŽ. Proposition A . Let Hp be the Sylow p-subgroup of H for some prime p. It suffices to show that Hp : N. p Observe that NHGpŽ.= OGŽ.by Proposition C. In particular, Hp is normalized by the Sylow q-subgroups of N for all q / p. Since Hp permutes with some Sylow p-subgroup of N by StatementŽ. b , we have

HNpps NH by StatementŽ. a . Without loss we thus may assume that H s Hp is a p-group. If S is a Sylow p-subgroup of G, we have HS s SH s S. So assume in what follows that S is a q-group for some prime q / p. H p Recall that G00s HN is a group and that H g SylŽ.G . Since OGŽ. p normalizes H and G s OGNŽ., we have HNGs H s 1. Hence we may assume that G s HN, and we have to show that G s N. Let P be a Sylow p-subgroup of G. Then H : P as H is subnormal in GH G. Furthermore, S permutes with P s HPŽ.l N . From S s S : HS G s SH we deduce that S l P is a normal subgroup of P contained in H. p G Since G s OGPŽ. and so HPGs H s 1, this implies that S l P s 1 G and S s S. Hence S is normal in G, as desired.

7. SYSTEM NORMALIZERS

In this section we assume that G is solvable. Let ⌺ be a Sylow system of G, that is, ⌺ consists of pairwise permutable Sylow subgroups of G containing just one for each prime dividing <

H THEOREM C. Let H g SylŽ.G . Then HrHG : ZGϱŽ.rHG if and only if H permutes with NŽ.⌺ .

Proof. One knows that NŽ.⌺ HGGrH is a system normalizer of GrHG whose normal core is just ZGϱŽ.rHG wx2, VI.11.3 and 11.11 . Thus the ‘‘only if’’ part is obvious. For the proof of the converse assume Ž.G, H is a counterexample with <

p Ž.1 HGps 1, H : O Ž G . for some prime p, and NG Ž H .= OGŽ..

Since NŽ.⌺ HGGrH is a system normalizer of GrHG, by minimality HG s 1Ž. so that H is nilpotent by Proposition A . From Proposition C and StatementsŽ.Ž. b , a it follows that the Sylow subgroups of H permute with NŽ.Ž⌺ cf. the proof of Proposition D . . Thus the choice of ŽG, H .gives assertionŽ. 1 . Let P ⌺ be the Sylow p-subgroup, and let R Ł S be the g s S g ⌺ RÄ P4 corresponding p-complement. Then N s P l NRGŽ.is the Ž unique . Sylow p-subgroup of NŽ.⌺ wx3, VI.11.2 . p Ž.2 G s OŽ. G N and HN s 1. The first statement follows from the fact that NŽ.⌺ covers each central chief factor of G. Apply thenŽ. 1 . Ž.3 wxH, R / 1. ByŽ. 1 and Lemma B, and the choice of ŽG, H ., H is not centralized by p x OGŽ.. If the commutator group wxH, R s 1 then we would have wH, R x p p p s 1 for all x g OGŽ., because NHG Ž .= OGŽ.. Now use that OGŽ.s RGpis generated by the OGŽ.-conjugates of R. Ž.4 P s HN. By hypothesis H permutes with NŽ.⌺ . Hence by Ž.Ž 1 and Statement Ž.. b

P00s HN is a subgroup of P. Let G s PR0; this is a subgroup of G since H is normalized by R and R is normalized by N. AssumeŽ. 4 is false so that G0 is a proper subgroup of G. Consider the Sylow system ⌺00of G obtained from ⌺ by replacing P by P0. The corresponding system normalizer NŽ.⌺ of G contains NŽ.⌺ . In fact, N P NRŽ.is the Sylow 00 s 0l G 0 p-subgroup of NŽ.⌺00as well. Since the p-complement of NŽ.⌺ normalizes H byŽ. 1 , H permutes with NŽ.⌺0 . Moreover, byŽ. 2 we have H H 1. As the theorem holds for G , H, NŽ.⌺ , from Lemma B GN0 s s 00 p we get that H is centralized by OGŽ.0 . But this contradictsŽ. 3 as p R : OGŽ.0 . p Ž.5 OGŽ.: HR s RH. PERMUTABLE SUBGROUPS 293

p GPH We have OGŽ.s R s R s R , because G s PR, P s NH byŽ. 4 , H and N normalizes R. Clearly HR s RH and R : HR. p Ž.6 wOGpŽ., OGŽ.x : H. This is immediate fromŽ.Ž. 1 , 5 since the commutator group is contained p in OGpŽ.l OGŽ.: P l HR s HPŽ.l R s H. p Conclusion. Of course, wOGpŽ., OGŽ.x is a normal subgroup of G.By Ž.6 and Ž. 1 it must be trivial. It follows that wxH, R s 1, which is in contrast toŽ. 3 . The proof is complete.

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