Autocommutator Subgroups of Finite Groups

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1 JOURNAL OF ALGEBRA 90, ARTICLE NO. JA96692 Autocommutator Subgroups of Fiite Groups Peter V. Hegarty Departmet of Mathematics, Priceto Uiersity, Priceto, New Jersey Commuicated by Gordo James Received July 9, 996 We preset some ew results about the absolute ceter ad autocommutator subgroup of a group, cocepts which were itroduced i a earlier paper. 997 Academic Press. INTRODUCTION I 4, we itroduced the otios of the absolute ceter LG ad the autocommutator subgroup G* of a group G, ad proved the followig basic result:.. THEOREM. If GL G is fiite the so are G* ad Aut G. May atural questios ow beg to be asked. The most obvious is whether aythig like a coverse to Theorem. holds. We ca give a complete aswer, as follows:.2. THEOREM. If G* ad AutŽ G. are both fiite, the so is GLŽ G.. Howeer, there exists a group G for which G* is fiite but GLŽ G. is ifiite. Proof. First suppose that both G* ad AutŽ G. are fiite. Let AutŽ G.. The ŽG: C Ž.. is fiite, sice G* is fiite. But LG G C Ž. AutŽG. G, ad sice this is a fiite itersectio, it follows that GLG is fiite. If oe removes the hypothesis that AutŽ G. be fiite, the result breaks dow. I 3, Fourelle costructs, for every odd prime p, a ifiite group G for which AutŽ G. is a ucoutable elemetary abelia p-group. For these groups, oe checks readily that G* is fiite, but GLŽ G. is ifiite..3. Remark. If G is fiitely geerated, ad G* is fiite, the AutŽ G. is clearly fiite. So GLŽ G. is also fiite by our theorem. I fact, G is a fiite cetral extesio of a cyclic group, by Alperi s result $25.00 Copyright 997 by Academic Press All rights of reproductio i ay form reserved. 556

2 AUTOCOMMUTATOR GROUPS 557 The ext issue is whether the result of Theorem. ca be stregtheed. Specifically, oe ca ask whether there exists ay ifiite group G such that GLŽ G. is fiite. A defiitive aswer is cotaied i the followig result:.4. THEOREM. Let p be a odd prime. The, for eery 4, there exists a ifiite group G with GL G elemetary p-abelia of rak. Proof. Agai we refer the reader to the paper of Fourelle 3. He costructs, for every prime p, ifiite groups G for which AutŽ G. is elemetary p-abelia of fiite rak. A straightforward aalysis of these groups, for p odd, yields our theorem. We have bee able to prove that if, 2, or 3, the above theorem is false Žoly the case 3 causes ay problemsthe proof is tedious rather tha difficult; it is therefore omitted, ad may be obtaied from the author if desired.. It is also false for p 2 ad ay, as oe may easily show. I the oabelia case, it is easy to see, for example, that if GLŽ G. is ay dihedral group the G must be fiite. However, we do ot curretly possess ay systematic treatmet of the problem of determiig those fiite groups K for which there exists a ifiite G such that GLŽ G. K, eve i the case where K is abelia. So for the remaider of the paper we shall tur to discuss other problems. 2. MAIN RESULTS The results of Sect. may be tatalisig, but ot very rewardig. So let us chage focus. The followig result is well kow Žsee, for example. 2 : 2.. THEOREM. Let K be ay fiite group. The there are oly fiitely may fiite groups G such that AutŽ G. K. Ideed, oe ca write dow explicitly a fuctio g: N N such that the followig holds: If p is ay prime gž h. h ad G p p the AutŽ G. p p. For example, oe ca take gž h. h 2 6, for h 3. We shall ow provide results similar to Theorem 2., with AutŽ G. replaced by either GLŽ G. or G*. The case of GLŽ G. is by far the simplest: 2.2. THEOREM. Let K be a fiite group. The there are oly fiitely may fiite groups G for which GL G K. Proof. Let K ad suppose G is fiite with GLŽ G. K. By Theorem 2. of 4, there is a fuctio f: N N such that AutŽ G. fž.. This, combied with Theorem 2., yields the result.

3 558 PETER V. HEGARTY 2.3. Remark. From this proof it is clear that, as i Theorem 2., oe could write dow explicitly a fuctio g: N N such that if G is fiite ad GLŽ G., the G gž.. However, sice the boud i Theorem 2. of 4 seems extravagat, this g is ulikely to be the best possible Ž see Sect. 2. of this paper Remark. As is well kow, the first part of Theorem 2. is false if we allow G to be ifiite. The same is true of Theorem 2.2 ad ideed of Theorem 2.5 below. Oce agai, the groups costructed i Fourelle s paper 3 illustrate all three of these facts. We ow tur to the mai result of this paper THEOREM. Let K be a fiite group. The there are oly fiitely may fiite groups G for which G* K. As the proof will show, this is essetially a result about p-groups. We begi with five lemmas, which costitute the basic tools eeded for the proof LEMMA. Let K be ay fiite group, d 0 a fixed positie iteger. The there are oly fiitely may d-geerated fiite groups G for which G* K. Proof. Suppose G is a d-geerated fiite group for which G* K. We shall show that AutŽ G. is bouded. This will immediately yield the lemma, because of Theorem 2.. So let x, x,..., x geerate G, ad let AutŽ G. 2 d. is completely determied by specifyig the set x 4 i xi i d of elemets of G. But all these elemets lie i G*, by defiitio of the latter. So AutŽ G. is certaily bouded by the umber of fuctios from the set,..., d4 K, i.e., by K d. The lemma follows. The followig crucial result must surely appear i some form i the literature, though I have ot bee able to locate it. 2.7 LEMMA. Let p be ay prime, N ay fiite p-group. Let deote the class of all fiite p-groups which possess a ormal subgroup, which is isomorphic to N, ad such that the factor group is abelia. The there exists N Ž idepedet of p. N such that the followig holds: If G satisfies dž GN. the C Ž N. ZŽ N. N G cotais a direct factor of GN. Explicitly, we may take N bžb.2, where b log ŽN.. N Proof. Ž This is essetially just the pigeohole priciple.. Let G ad suppose dgn, with as above. Defie a sequece N of N N k p

4 AUTOCOMMUTATOR GROUPS 559 subgroups of N as follows: N G, N ; N G, N for k 0. Ž. 0 k k bk Sice N G, we have that N p for k b ad that N. 4 k b Also, we defie a sequece Cr of positive itegers, accordig to the followig rule: C ; C N r.c for r. Ž 2. r r Note that C. Now let x,..., x 4 b N i d be a set of elemets of G which geerate G mod N ŽddGN.. Sice dn bwe ca choose elemets,..., 4 b of N which geerate it. Fix ay such choice. Now we have d fuctios :,..., b4 N defied by, j 0 Ž t. x, Ž jd,tb.. Ž 3., j j t But there are certaily o more tha N b distict fuctios from,..., b4 to N. Sice dgn C, there is a subset of S of,..., d4 0 b such that Ž S. C ad for all j, j S Ž pigeohole priciple!. b, j, j. It follows that xx,n j j N for all j j S. I other words, there exists a set y,..., y 4 C b of elemets of G which are idepedet mod N, geerate a direct factor of G mod N, ad satisfy y, N j N for j,...,c. Now we ca defie fuctios :,..., b4 N Ž j,...,c. b 2, j b by Ž t. y, Ž 4. 2, j j t But there are certaily o more tha N b distict fuctios from,...,b4 to N so, reasoig as above, we coclude that there exists a set z,..., z 4 C b of elemets of G which are idepedet mod N, geerate a direct factor of G mod N, ad satisfy z, N j N2 for j,...,c b. It is clear how this argumet is goig to proceed. Sice N 4 b ad C we will evetually be able to coclude that there exists w G which geerates a Ž cyclic. direct factor of G mod N ad satisfies w, N. This proves Lemma 2.7. The followig stroger result follows, with some care, from repeated applicatio of Lemma LEMMA. Let p, N, ad be as i Lemma 2.7. The there is a fuctio f : N N, with f Ž. N N N, such that the followig holds: If G satisfies dž GN. f Ž i. N the there is a subgroup H of G with the followig properties: Ž. HN. Ž. 2 H is geerated mod N by elemets of ZŽ H.. Ž. 3 HN cotais a direct factor of GN of rak i.

5 560 PETER V. HEGARTY The ext tool we require is a simple homological result, a proof of which may be foud i 5, 7., Satz I LEMMA. Let G be ay group ad N a ormal subgroup. The the set of automorphisms of G which cetralize both N ad GN form a group which is caoically isomorphic to Z ŽGN, ZŽ N... For the ext lemma, Ž G. deotes the set of primes dividig the order of a fiite group G ad Ž G. deotes the umber of such primes LEMMA. Let K be a fiite group ad let K deote the class of fiite groups G such that G K. The eery G K possesses a ormal subgroup N with the followig properties: Ž. N is ilpotet ad characteristic i G. Ž. 2 G: N is bouded by a fuctio of K ad G oly. Proof. The proof is by iductio o K. The result is trivial for K, so suppose it true for K. Now let K have order ad cosider G K.If Gis itself ilpotet we may take N G. Otherwise, let N be the itersectio of the ormalizers of all Sylow p-subgroups of G ŽpŽ G... Evidetly, N is characteristic i G ad Ž G : N. is bouded by some fuctio of K ad Ž G.. But equally clearly oe sees that N G is properly cotaied i G. So we ca apply our iductio hypothesis, ad the lemma follows immediately. Proof of Theorem 2.5. We suppose that Theorem 2.5 is false for some Ž heceforth fixed. fiite group K, ad we obtai a cotradictio. I what follows O Ž G. deotes the maximal ormal -subgroup of a fiite group G, where is ay set of primes. Sice our theorem is false for K we ca fid a ifiite sequece G of fiite groups such that G K. Each G is a semidirect product of O Ž G. by a abelia Ž K. Ž K. -group, say A. But, for each, AutŽ G. is a extesio of a Ž K.-group Ž amely, Z ŽG G, ZG Ž.., by Lemma 2.9. by a subgroup of AutŽ K.. It follows, from Theorem 2., that the order of each A is bouded. So we ca fix a choice A of a abelia Ž K. -group ad suppose, without loss of geerality, that each G has Ž K.-complemet A. The poit is that Ž G. is ow the same for each. To ease otatio, write J for O Ž G. Ž K.. For p p K we also write T for the subgroup of G geerated by G ad the p-part of G G. Combiig Lemmas 2.6 ad 2.0, we ca ow coclude that, for some p Ž K. ad K0 a p-subgroup of K, the followig holds: Without loss of geerality, the sequece G may be chose so that Ž. G is the same for all.

6 AUTOCOMMUTATOR GROUPS 56 Ž. 2 Each J cotais a characteristic p-subgroup P such that P G K ad dppg Ž. 2f Ž. 0 K, where fk is the positive-iteger 0 0 valued fuctio of Lemma 2.8. Ž. Ž p. Ž p 3 Also, dt G f ad T : P. is the same for all. K0 Sice the factor group GG* is abelia for ay group G, we may apply Lemma 2.8 to the groups P. A techical poit arises here, which will become importat later. So, for the momet, please just accept what I am about to say. For each, we ca list the ivariats of P P G ad divide them ito two disjoit subsets U ad V, each cotaiig at least f Ž. K 0 umbers, such that o umber i U is bigger tha ay umber i V. The, by Lemma 2.8, there exists, for each, a characteristic subgroup H of P with the followig properties: Ž. HPG. Ž. 2 H is geerated mod P G by elemets of ZŽ H.. Ž. 3 HPG is a direct factor of PP G of rak. Ž. 4 All the ivariats of H P G come from the set U. Now, for each, fix the followig otatio: A Z P G Ž., BZŽ H., CBA, GH. Ž 5. The we have a short exact sequece of -modules 0 A B C 0 Ž 6. ad, hece, a log exact sequece of homology groups 0 H 0 Ž, A. H 0 Ž, B. H 0 Ž, C. H Ž, A. H Ž, B. H Ž,C. H 2 Ž, A.. Ž 7. It follows easily, ad i a purely formal maer, from Ž.Ž 7 just write dow what exactess meas at each step ad take it from there. that ž / H Ž, C. Z Ž, B. H Ž,C. H Ž, A. H Ž, A. Z Ž, A.. Ž 8.

7 562 PETER V. HEGARTY I claim that, for sufficietly large, the bracketed term will be greater tha. Admit this for the momet. The, for some, Z Ž, B. Z Ž, A.. By Lemma 2.9, this is equivalet to the existece of a automorphism of G which cetralizes both H ad GH, but ot G P G. But this cotradicts the very defiitio of G. So, i order to complete the proof of Theorem 2.5, it suffices to Ž p establish our claim above. To simplify otatio, let d dt G. ad d. First, for ay, simple recourse to the defiitios gives 0 H Ž, A. A K, Ž H Ž, A. A K. Ž 0. Next, we ote that C is a trivial -module Ž sice G* G for ay group G.. Hece, H Ž, C. HomŽ, C. ad H 0 Ž, C. C. It is here that the techical poit referred to earlier becomes importat. Property Ž. 4 of the group H guaratees Ž a fact of which oe easily covices oeself. that the quotiet HomŽ, C.C grows at least as rapidly, with icreasig, as p. Hece, the bracketed term i Ž. 8 grows at least as rapidly, with icreas- Ž ig, as pk 2.. Our claim follows immediately, ad the proof of Theorem 2.5 is complete. 2.. OPEN PROBLEMS. Ž. a Usig our proofs of Theorems 2.2 ad 2.5, oe could compute explicit bouds for the order of a fiite group G i terms of GLŽ G. ad G*, respectiely. Howeer, these bouds will be ery extraagat. So it is a alid questio to ask for best possible, or ee reasoable, bouds, like those gie i Theorem 2.. Ž b. Gie a systematic treatmet of the problem, raised earlier, of determiig those fiite groups K for which there exists a ifiite group G such that GLG K. More realistically, fid the smallest such K ŽTheorem.4 4 implies that K REFERENCES. J. L. Alperi, Groups with fiitely may automorphisms, Pacific J. Math. 2 Ž 962., T. Exarchakos, Automorphisms of a fiite p-group, Caad. J. Math. 32 Ž 980., T. A. Fourelle, Elemetary abelia p-groups as automorphism groups of ifiite groups II, Housto J. Math. 9 Ž 983., P. V. Hegarty, The absolute ceter of a group, J. Algebra 69Ž.Ž , B. Huppert, Edliche Gruppe, I. Spriger, New York, 967.