On the Fixed-Point Set and Commutator Subgroup of an Automorphism of a Group of Finite Rank

On the Fixed-Point Set and Commutator Subgroup of an Automorphism of a Group of Finite Rank

B. A. F. WEHRFRITZ

ABSTRACT- Le tfbean automorphism of a groupG. ForGpolycyclic, Endimioni and Moravec in [1] discuss the relationship between the fixed-point setCG(f) and the commutator subgroup [G,f] offinG. Here we extend these results to soluble groups satisfying various rank restrictions.

Supposefis an automorphism of thepolycyclic groupG. Endimioni and Moravec in [1] prove the following.

i) IfCG(f) is finiteandfhas order 2, then [G,f]⁰is finite.

ii) IfC_G(f) is finite, then so isG=‰G;f].

iii) Ifjfjis finite, then so is the index (G: [G,f].CG(f)).

Of course C_G(f) denotes the set of fixed points of f in G, [G,f] = hg ¹:gf:g2Giand [G,f] is af-invariant normal subgroup ofG. Here we generalize these results to groups with some sort of rank restriction.

Generally our groups at least have finite Hirsch number and satisfy something weaker than the FAR condition. We must start by defining these terms.

If a groupGhas a series of finite length each factor of which is either locally finite or infinite cyclic, then the number of infinite cyclic factors in such a series is an invariant ofG, which wecall theHirsch number ofG(it is also sometimes called the torsion-free rank ofG). This is theweakest of the rank restrictions considered here. The structure of groups with finite Hirsch number is discussed in many places (e.g. see [3]). From our point of view the most convenient description of the structure of such a group is

(*) Indirizzo dell'A.: School of Mathematical Sciences, Queen Mary University of London, MileEnd Road, London E1 4NS, England.

E-mail: [email protected]

given by Lemma 4 of [6]. A group satisfies min-pfor theprimepif itsp- subgroups satisfy theminimal condition on subgroups. An FAR group (that is, a finite-abelian-ranks group, also called anSo group) is a soluble group with finite Hirsch number that satisfies min-pfor every primep. Se e for example [3] for a discussion and alternative definitions of the class of FAR groups. In particular all solublegroups of finite(PruÈfer) rank are FAR groups. The following summarises our main conclusions in this paper.

THEOREM. Let f be an automorphism of the finite extension G of a solubleFARgroup. Then the following hold.

i) If CG(f)is finite and f²ˆ1, then[G,f]⁰is finite.

ii) If CG(f)is finite, then G/[G,f]is finite.

iii) Iffhas finite order, then the index(G: [G,f].C_G(f))is finite.

In fact we can weaken the FAR hypothesis in the theorem considerably in many places. For i) it suffices to assume only thatGis a group with finite Hirsch number (even in a weak sense). For iii) it suffices to assume thatG is a soluble-by-finite group with finite Hirsch number satisfying min-pfor all primes p dividing theorder of f. However we do need some min-p hypothesis; suppose Gis thedirect product of thecyclic groups ha_iji of orderp fori1, jˆ1, 2 andp any (fixed) prime. Definef :G!G by a_i1fˆa_i1anda_i2fˆa_i1a_i2for eachi1. Thenfis an automorphism ofG of primeorderpwith‰G;fŠ ˆC_G…f† ˆ ha_i1:i1i. Thus…G:‰G;fŠ:C_G…f††

is infinite. TriviallyG has Hirsch number 0. As Endimioni and Moravec point out in [1], we do needfof finite order in iii), even ifGis free abelian of rank 2. Notethat iii) implies thatG=‰G;fŠis isomorphic to a finite extension of a section ofCG(f); see Remark 6 below.

We are unable to weaken the FAR condition in ii) in general, though we can in special cases. Note that ii) implies thatCS(f) is finite for everyf- invariant sectionSofG. Iffhas finiteorder (withC_G(f) finite) it suffices for ii) to hold to assumethatGis just soluble-by-finite with finite Hirsch number, see Remark 7. If f has prime order, then it suffices simply to assumethatG has finite Hirsch number, since Theorem 1 of [6], for ex- ample, reduces the problem to the soluble-by-finite and hence previous case. A (torsion-free)-by-finite group with finite Hirsch number is always a finite extension of a soluble FAR group, so this case is also covered. Of courseif G satisfies the conclusion of iii) andCG(f) is finite, then again G=‰G;fŠis always finite.

If in ii) theorder offis infinite, then we do need some sort of min-p condition, even if f acts fixed-point freely on G. For e xample , le t G

denotethecartesian product of theadditivecyclic groups haiiof order pⁱ, whe re pis any primeandiˆ1;2; ::::. De fine f:G!Gby…eiai†fˆ ……e_i‡pe_{i 1}†a_i†, whe re the e_i are integers with e0ˆ0. Then f is easily seen to be an automorphism ofGof infiniteorder withCG(f† ˆ h0iand [G,fŠ ˆpG. In particular G=‰G;fŠ is infiniteand trivially G is abelian with Hirsch number 0. Clearly Gis uncountable, but if DG denotes thedirect product of thehaii, the n Hˆ S

i1Df ⁱ gives a countable counter example. Finally the converse of ii) is false, even if G is poly- cyclic and fhas primeorder, seeExample8 below.

By removing from our proof of the above theorem the complications caused by the presence of finite rank but infinitely generated sections, one can use our methods to construct somewhat shorter proofs of i), ii) and iii) than those given in [1]; this is especially the case with iii).

LEMMA1. Let G be a group with a series of finite length, each factor of which is either periodic or infinite cyclic. Iffis an automorphism of G with CG(f)finite andf²= 1,then‰G;fŠ⁰is finite.

Part i) of the Theorem follows at once from this. Note that the condition onGin Lemma 1 is weaker than having finite Hirsch number as we have defined it in the introduction above.

PROOF. NowGhasf-invariant normal subgroupsABwith A and G/

B finite and B/A abelian and inverted by f, see [5] Theorem 2. Also

‰G;f;BŠ A; this follows from [1] Lemma 1, but it also follows immediately from simply expanding the commutator‰g;f;bŠfor any g2Gandb2B.

Then‰G;fŠA=Ais centre-by-finite. Hence‰G;fŠ⁰A=AandAareboth finite.

Consequently‰G;fŠ⁰is finite.

LEMMA 2. Let A be an additive torsion-free abelian group of finite rank. Iffis an automorphism of A with C_A…f† ˆ h0i, then A=A…f 1†is finite.

PROOF. Clearlyf 1 extends to an automorphismuofV ˆQZA. In particular detu6ˆ0. By the Cayley-Hamilton Theorem detu2Q‰uŠu End_QV. Thus there is a positive integerewithe2Z‰uŠuˆEnd_QV. He nce eAA:Z‰uŠuAuˆA…f 1†and thereforeA=A…f 1†is finite.

For any group G denotetheuniquemaximal locally finitenormal subgroup ofGbyt…G†.

LEMMA3. Let G be a group with finite Hirsch number h and witht…G†

finite. Iff is an automorphism of G with CG(f) finite, then G=‰G;fŠis finite.

PROOF. We induct on h; clearly ifhˆ0 theclaim is vacuous. Sup- poseh1. Then Ghas a non-trivial torsion-freeabelian characteristic subgroupAwith G/A(torsion-free)-by-finite (e.g. see Lemmas 4 and 6 of [6]). Clearly C_A…f† ˆ h1i, so by Le mma 2 we have …A:‰A;fŠ† ˆe finite. SetBˆA^e. The nBis characteristic in G, B ‰G;fŠand A=Bis finite. Further the map a7!a^e is an isomorphism of A onto B com- muting withf, so…B:‰B;fŠ† ˆ …A:‰A;fŠ† ˆe. By [4] Le mma 1 we have jC_G=B…f†j ejCG…f†j<1. By induction on h thegroup G=‰G;fŠB is finite. SinceB ‰G;fŠwehaveG=‰G;fŠ finite.

LEMMA4. Let G be a finite extension of a soluble FAR group. Iffis an automorphism of G withjC_G…fj ˆn finite, then G=‰G;fŠis finite.

Part ii) of the Theorem follows at once from Lemma 4.

PROOF. Se tpˆ{primesp:pn} andTˆt…G†. By [2] 3.17 there is a characteristic series

h1i ˆT0T1::::TsTG

of Gwith each T_i=T_{i 1} an abelianp⁰-group fori<s, T_s=T_{s 1} a divisible abelianp-group andT=T_sfinite.

Suppose AˆT₁ is a p⁰-group. Clearly C_A…f† ˆ h1i. Then the map g:a7!a ¹:af is an isomorphism of A into itself. If Ap is the p-primary component of A, thenApApg. IfAp>Apg, then Apgⁱ>Apg^i‡1 for all i0. But A_p satisfies the minimal condition (G satisfies min-p for all primesprecall). ThereforeA_pˆA_pgfor allpandAˆAg ‰G;fŠ. Also if K=AˆC_G=A…f†, then Lemma 1 of [4] yields that…K:A† n.

Now supposetheAˆT1is a divisibleabelianp-group (thecasesˆ1).

Then Ag is divisible and hence is a direct summand of A. But kergj_AˆC_A…f†is finitesoAAgand alsoAis a direct product of a finite (invariant) number of PruÈfer groups. ThereforeAgˆA. Again weobtain …K:A† n.

A simpleinduction on s shows thatTs ‰G;fŠand that thecentralizer of f on G=Ts has order at most n. Further t…G=Ts† is finite. Therefore G=‰G;fŠTs is finite by Lemma 3. ConsequentlyG=‰G;fŠis finite.

LEMMA5. Let G be a soluble-by-finite group with finite Hirsch number andfan automorphism of G of finite order m. Suppose G satisfiesmin-p for every prime p dividing m. Then…G:‰G;fŠC_G…f††is finite.

Part iii) of the Theorem follows from Lemma 5.

PROOF. If N is a f-invariant subgroup of G of finiteindex with …N:‰N;fŠC_N…f††finite, then…G:‰G;fŠC_G…f††is also finite. In particular we may assumethatGis soluble. SetC=C_G(f).

Supposefirst thatTˆt…G†is finite. Weprovethis caseby induction on theHirsch number of G. If Gis finitetheconclusion is vacuous. If not G has a non-trivial torsion-free abelian characteristic subgroup A withG/A(torsion-free)-by-finite ([6] Lemmas 4 and 6 again). Define the map g:G!G by g7!‰g;fŠ and let c denotetheendomorphism 1‡fj_A‡ …fj_A†²‡::::‡ …fj_A†^{m 1} of A. Clearly c…gj_A† ˆ0, so AcC.

Also if a2A, the n a^m is congruent to ac modulo Ag. The re fore A^mAg:Ac ‰A;fŠ…A\C†:

LetA^mKGwithK=A^mthe centralizer inG=A^moff. NowA=A^m is finitesinceA has finiterank, so induction applies to G=Am. Conse - quently wemay assumethat …G:‰G;fŠK†is finite. Also …A\K:A^m†is finiteand Kˆ …A\K†C, the latte r by Le mma 15 of [6]. The re fore …K:A^mC†is finite; thus…G:‰G;fŠA^mC†is finite. ButA^m ‰A;fŠ…A\C).

Consequently …G:‰G;fŠC† is finite. This completes the proof of the T- finitecase.

Let pdenote the set of prime divisors of m. By [2] 3.17 there is af- invariant normal series h1i ˆT0T1::::Ts TofGas in theproof of Lemma 4, with factorsT_i=T_{i 1}abelianp⁰-groups or divisibleabelianp- groups andT=T_s finite. For each ithesectionfx2T_i:x^m2T_{i 1}g=T_iis finite. Hence by our initial remark we may assume that each of these sections is central in G. By theT-finitecasewemay assumethat G=Ts

satisfies theconclusion of thelemma. Thus by induction on s wemay assumethat G has a periodic abelian characteristic subgroup A with A^mˆA, Bˆ fa2A:a^mˆ1g finiteand central in G and …G:‰G;fŠK†

finite, whereKis given byK=AˆC_G=A…f†.

Letk2K. Thenkg2AˆA^m, saykgˆa^mfora2A. Nowa^m2acAg andacgˆ1,cbeing as in theT-finitecase. Thusa^mˆc:dgforcˆac2C and somed2A. Hencekgˆc:dgand

…kd ¹†gˆd:kg:d ¹fˆd:c:dg:d ¹fˆc2A\C:

Notethat theargument also implies that AˆAg…A\C† ˆ

‰A;fŠ…A\C†. Sincetheactions ofkandf onAcommute, so kandkd ¹ normalizeA\C. Thenfstabilizes the series

h1i A\C hkd ¹i…A\C†

and hence ‰kd ¹;fŠ^mˆ ‰kd ¹;f^mŠ ˆ1. Thus kd ¹2L, where L=BˆC_G=B…f†. Hence KLAˆAL; indeed KˆAL. Also f stabilizes the series h1i BL and B is central in L. Therefore gj_L is a homo- morphism ofLintoBwith kernalC. Thus…L:C† jBj<1. Consequently …K:AC† is finiteand hence…G:‰G;fŠAC† is finite. As we saw above Aˆ ‰A;fŠ…A\C†. Therefore…G:‰G;fŠC† ˆ …G:‰G;fŠ‰A;fŠC is finite, as required.

REMARK6. SupposeGandfareas in Lemma 5. Then alwaysG=‰G;fŠis a finite extension of some section ofCG(f). For ifNdenotes the intersection of theconjugates inGof‰G;fŠC_G…f†, thenG=Nis finiteby Lemma 5 and

‰G;fŠ N. ThusNˆ ‰G;fŠC_N…f†andN=‰G;fŠ C_N…f†=…‰G;fŠ\C_N…f††.

In particular if X is a subgroup and imageclosed class of groups containing C_G(f) and if F denotes the class of finite groups, then G=‰G;fŠ 2XF. IfGis polycyclic, this is Proposition 1 of [1].

REMARK 7. Let G bea soluble-by-finitegroup with finiteHirsch number. Iffis an automorphism ofGof finiteorder withCG(f) finite, then G/[G,f] is finite.

PROOF. Se t Tˆt…G†. Repeated use of [6] Lemmas 9 and 13a) yield thatT=‰T;fŠandC_G=T…f†are both finite. ThenG=‰G;fŠTis finiteby Lemma 5 (or Lemma 3 if you prefer). Trivially‰T;fŠ ‰G;fŠ. ThereforeG=‰G;fŠis finite.

EXAMPLE8. Letvbea primitivep-th root of unity inCfor someprime pand putRˆZ‰vŠ CandGˆTr₁…3;R†. (HereZdenotes the integers, C the complex numbers and Tr₁…3;R† thelower unitriangular group of degree 3 over the ringR; laterQwill denote the rational numbers.) ThenG is a 2…p 1†-generator nilpotent group of class 2 (and in particular is polycyclic). IfZdenotes the centre ofGand {eij} thestandard set of matrix units inC³³, thenZˆ1‡Re31ˆG⁰andZRandG=ZRR.

Let f denote the automorphism of G induced by conjugation by the diagonal matrix diag…1;v;1†. Thenf has orderp, centralizes Zand acts

fixed-point freely on G=Z. If xˆ1‡ae21 and yˆ1‡be32 for some a, b2R, then a simple calculation shows that

‰x ¹:xf;y ¹:yfŠ ˆ1‡ …v 1†²v ¹abe₃₁2 ‰G;fŠ⁰: Also 1‡v‡v²‡::::‡v^{p 1}ˆ0, so pˆ P

1i<p…1 vⁱ† 2 …v 1†R.

Therefore 1‡p²Re31Z\ ‰G;fŠ⁰ and in particular …‰G;fŠZ:‰G;fŠ† ˆ …Z:Z\ ‰G;fŠ†dividesp^{2…p 1†}:

Supposegˆ …g_ij† 2G. Themapg7! …g21;g32†is a homomorphism ofG onto RR with kernel Z that maps ‰G;fŠ onto …v 1†…RR†. Thus …G:‰G;fŠZ†also dividesp^{2…p 1†}. ConsequentlyG=‰G;fŠis finiteof order dividingp^{4…p 1†}. HoweverCG…f† ˆZ, which is infinite. Thus the converse of Part ii) of the Theorem does not hold, even for polycyclic groups and au- tomorphisms of finiteprimeorder.

Supposeinstead of Tr₁…3;R†wesetGˆTr₁…3;K†, whereKis thefield Q‰vŠ C, but withfstill given by conjugation by diag…1;v;1†. Then here Gis torsion-free nilpotent of finite rank and class 2 andZˆC_G…f† Kis infinite,Zbeing the centre ofGagain. Our previous argument shows that G=‰G;fŠ has finite exponent (dividing p³). But Gis divisible, so here we havejfj ˆp,Gˆ ‰G;fŠandCG(f) infinite.

REFERENCES

[1] G. ENDIMIONI- P. MORAVEC,On the centralizer and the commutator subgroup of an automorphism, Monatsh. Math., to appear.

[2] O. H. KEGEL- B. A. F. WEHRFRITZ,Locally Finite Groups, North Holland Pub. Co., Amsterdam 1973.

[3] D. J. S. ROBINSON,Finiteness Conditions and Generalized Soluble Groups (2 vols.), Springer-Verlag, Berlin 1972.

[4] B. A. F. WEHRFRITZ, Almost fixed-point-free automorphisms of soluble groups, J. Pure& Appl. Algebra,215(2011), pp. 1112-1115.

[5] B. A. F. WEHRFRITZ, Almost fixed-point-free automorphisms of order 2, Rend. Circ. Mat. Palermo,60(2011), pp. 365-370.

[6] B. A. F. WEHRFRITZ,Almost fixed-point-free automorphisms of prime order, Cent. Eur. J. Math.9(2011), pp. 616-626.

Manoscritto pervenuto in redazione il 5 settembre 2011.