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Subgroups Permutable with All Sylow Subgroups JOURNAL OF ALGEBRA 207, 285᎐293Ž. 1998 ARTICLE NO. JA987429 Subgroups Permutable with All Sylow Subgroups Peter Schmid Mathematisches Institut, Uni¨ersitatÈÈ Tubingen, Auf der Morgenstelle 10, D-72076, Tubingen,È 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 . 285 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 situa- tion 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 normal- izes 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 sub- groups which cover each central chief factor of the underlying solvable group and avoid the eccentric ones. Permutability with all system normaliz- ers 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-sub- group 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 <<G q <G : H <minimal. Then HG s 1. Put x D s F ² H, H :. xgGRNHGŽ. x G Clearly H : D and each ² H, H :Ž.g Syl G H . The normal closure D s G x H and the normal core D ² H, H :G. In case D H Gxs F g G R NGŽ H . s G the result would follow, by our choice of Ž.G, H , because then H s G D rDG.So H is a proper subgroup of D. There is a prime p and a p-element x g G R NHGŽ.. Let P be a Sylow p-subgroup of G containing x. We have x H ; D : ² H, H :²: H, x :: HP s PH. We see that the index <<D : H is a p-powerŽ. for the chosen prime p , and we may conclude that H is normalized by the join OGpŽ.of all pЈ- G P elements of G.
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