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Home , Lattice (group), Lattice of subgroups

J. Theory 12 (2009), 859–872 Journal of DOI 10.1515/JGT.2009.018 ( de Gruyter 2009

The index problem

Stewart Stonehewer and Giovanni Zacher (Communicated by E. I. Khukhro)

Abstract. Given a group G and X d Y, with Y of finite index in X, then in general it is not possible to determine the index jX : Yj simply from the lattice ‘ðGÞ of subgroups of G. For example, this is the case when G has prime . The purpose of this work is twofold. First we show that in any group, if the indices jX : Yj are determined for all cyclic subgroups X, then they are determined for all subgroups X. Second we show that if G is a group with an ascending normal series with factors locally finite or abelian, and if the Hirsch length of G is at least 3, then all indices jX : Yj are determined.

1 Introduction A complete algebraic lattice L is called a subgroup lattice if there exists a group G and an from L to ‘ðGÞ, the lattice of all subgroups of G. Of course L deter- mines G to within a projectivity. We recall that the class Lg of subgroup lattices has been characterized in [15] (see [8, Theorem 7.1.17]). It is properly contained in the class of complete algebraic lattices, as easy examples show. Given L a Lg, a central problem in the study of groups, from a lattice-theoretical point of view, is to determine properties of groups G with ‘ðGÞ G L. These are prop- erties that are embedded in the structure of L and are invariant under projectivities. Write L ¼ ‘ðGÞ and let ðX; YÞ a L L. We consider the map n from L L to fN; 0; lg defined by 8 < 0ifY G X, ðX; YÞ 7! : l if Y c X and jX : Yj¼l, n if Y c X and jX : Yj¼n (finite).

Then we ask the following question: To what extent can the map n be determined simply from L? While the elements of L L, for which the values of n are 0, 1 or

The first author acknowledges support from GNSAGA and hospitality from the University of Padova. The second author acknowledges hospitality from the University of Warwick. 860 S. Stonehewer and G. Zacher l, are always apparent (see [8, (6.10)]), for other pairs ðX; YÞ with X > Y, this might not be the case. This happens, for example, when there is a singular projec- tivity between two groups (see [8, (4.2)]). Given L a Lg, determining the map n will be called the subgroup index problem for L. Suppose that H; K a L and H d K. We denote by ½H=K the lattice of subgroups between H and K. If we can find the values of n on ½H=K½H=K through a proce- dure inside L, based only on the lattice structure of L, then we say that the interval ½H=K has determined indices and that ½H=K is a d.i. interval. We say that ‘ðGÞ has determined indices if we can find the values of n on the whole of L L. In this case we call ‘ðGÞ a d.i. lattice. This subgroup index problem has been considered for finite groups in [1]. Here we get more precise information for a large con- structed from the locally finite groups and the abelian groups. We shall give con- ditions su‰cient for a subgroup lattice to have determined indices. We use subgroup lattice terminology and group terminology interchangeably whenever there is no am- biguity, for example, for an element of L ¼ ‘ðGÞ and a subgroup of G. We construct a class of groups starting from the locally finite groups and the abe- lian groups. Thus, using the notation of group classes and operations on classes in- troduced by Philip Hall (see [6, vol. 1]), we shall be working within the class

P´ nfLF U Ag¼D; consisting of all groups G with an ascending series fHlgld0 of normalS subgroups such that Hlþ1=Hl is locally finite or abelian for all l, and Hl ¼ m

the class D is lattice-closed, as we now proceed to show. Let G be a group and M c G. We say that M is a Dedekind subgroup of G, and write M cd G, if for all X; Y c G,

3M; Y4 V X ¼ 3M; Y V X4 whenever M c X and 3Y; M4 V X ¼ 3Y; M V X4 whenever Y c X:

If, for such a subgroup M, whenever 3M; g4 d Y d M with ½3M; g4=Y finite, it follows that j3M; g4 : Yj is finite for all g a G, then M is said to be permodular in G and we write M cD G. Thus the subgroups of prime order in Tarski groups are Dedekind, but not permodular. Clearly Dedekind subgroups are invariant under pro- jectivities. But so also are subgroups of finite index, by a result of Rips and Zacher [16]. Therefore permodular subgroups are also invariant under projectivities. A group G in which every subgroup is permodular is metabelian (see [8, Theorem 6.4.3]). If in The subgroup lattice index problem 861 addition G is non-abelian and non-periodic, then it is a periodic ex- tended by a torsion- of rank 1 (see [4]; see also [8, Theorem 2.4.11]). On the other hand, if G is non-abelian and periodic, then again its structure was de- scribed by Iwasawa (see [8, Theorems 2.4.13, 2.4.14]). In the light of these results, the fact that the class D is lattice-closed is perhaps now less surprising. In fact Schmidt proves in [8, Lemma 6.4.6] that if M is a non-trivial permodular subgroup of G and if all subgroups of M are permodular (in M), then MG (the normal closure of M) contains a non-trivial abelian of G. We can now prove our first result.

Proposition 1.1. Let G be a group and L ¼ ‘ðGÞ. Then GSa D if and only if L has an ascending chain of elements fHlgld0 such that H0 ¼ 1, l Hl ¼ G, each Hl cD G, and ½Hlþ1=Hl either is locally finite or has all its elements permodular (in Hlþ1) for all l.

Proof. If G a D, then by definition there is an ascending chain of the type described. Conversely, suppose that there is an ascending chain fHlgld0 as stated. We may assume that G A 1. It su‰ces to show that there is a non-trivial normal subgroup N of G such that N is either locally finite or abelian. We may assume that H1 is non-trivial and either is locally finite or has all its subgroups permodular. In the first G case, H1 is also locally finite, by [12, Corollary 2.4] (see also [8, Theorem 6.5.17]). G In the second case, as stated above, Schmidt proves in [8, Lemma 6.4.6] that H1 contains a non-trivial abelian normal subgroup of G. Thus the existence of N is established. r

If G a D, then we call the sum of all the torsion-free ranks of the abelian factors Hlþ1=Hl the Hirsch length of G and denote it by hðGÞ. It is an invariant of G.We shall prove two main results.

Theorem 1. Let G be a group and suppose that ½3g4=1 is a d.i. interval for all g a G. Then ‘ðGÞ is a d.i. lattice.

Theorem 2. Let G a D and suppose that hðGÞ d 3. Then ‘ðGÞ is a d.i. lattice.

In Lemma 3.5 we will prove that, with the hypotheses of Theorem 2, ½3g4=1 is a d.i. lattice for all g a G. Then Theorem 2 will follow from Theorem 1. The hypothesis hðGÞ d 3 in Theorem 2 is in fact necessary. For clearly we must have hðGÞ d 2. However, in [11, Theorem C] it is shown that there are torsion-free metabelian groups of Hirsch length 2 with non-index-preserving autoprojectivities. Also it is worth remarking that

the class D is not extension-closed.

Indeed let M ¼ MðZ; QÞ be the McLain group (see [6, vol. 2, pp. 14, 15]). Then M is generated by linear maps 1 þ aem;n, where a a Q and m; n a Z, m < n. Also there is 862 S. Stonehewer and G. Zacher an automorphism of M of infinite order defined by

1 þ aem;n N 1 þ aemþ1;nþ1:

Let G be the of M by the infinite generated by this automorphism. Since M is a Fitting group, it lies in the class D. However, any non- trivial normal subgroup N of G must intersect M non-trivially, and so contains an element 1 þ aem;n, a A 0 (see [6, Theorem 6.2.1 (iii)]). But then N contains the conju- gate element 1 þ aen;2nm, and hence contains

2 ½1 þ aem;n; 1 þ aen;2nm¼1 þ a em;2nm:

Therefore N cannot be abelian. Since G is torsion-free, G c D. Then forming the direct product of G with a free abelian group of rank 2 proves our claim. Our main objective in most of what follows will be to prove that cyclic intervals ½3g4=1 are d.i., given certain hypotheses. When 3g4 is infinite, then it is easy to see that this will be the case whenever we have a non-trivial subgroup 3h4 of 3g4 such that ½3h4=1 is d.i. Also, when 3g4 is finite, then its lattice is d.i. whenever we know jAj for each (i.e. minimal subgroup) A of 3g4. Similarly any cyclic interval ½3g4=1 will be d.i. if we know j3g4 : Mj for all maximal subgroups M of 3g4. Again when 3g4 is finite and 3h4 is a subgroup, then ½3g4=3h4 and ½3h4=1 d.i. imply the same for ½3g4=1. In Section 2 we prove Theorem 1 and consider hypotheses su‰cient to guarantee that all cyclic elements of lattices are d.i. In Section 3 we show that the class D sat- isfies these hypotheses (Lemma 3.5) and then Theorem 2 follows from Theorem 1. Our notation is standard. For a prime p and cardinal n d 2, Pðn; pÞ is the class of groups which are either an elementary abelian p-group of rank n or a semidirect product of an elementary abelian p-group A of rank n 1 by a group of prime order q (Ap) which induces non-trivial power automorphisms on A.Soq divides p 1. Also Pðn; pÞ is lattice-closed and features significantly in the theory of projectivities between finite groups (see [8, §2.2]). A group G is called a P-group if G a Pðn; pÞ for some p and n. When P is used to denote a product of lattices, then a direct product is always implied. A lattice is said to be irreducible if it is not the direct product of two non-trivial lattices. A minimal non-trivial element of a lattice is called an atom. Also H o G indicates that H is a maximal proper subgroup of G. The group of autopro- jectivities of G is denoted by PðGÞ. We write H z K for a semidirect product of a group H by a group K.IfH; K are subgroups of a group G, then CH ðKÞ and NH ðKÞ are, respectively, the centralizer and the normalizer of K in H. Also HG is the normal core of H in G. Finally, FðGÞ denotes the of G and ZðGÞ the centre of G.

2 Preliminary lemmas and proof of Theorem 1 In order to prove Theorem 1, we must consider cyclic intervals above permodular subgroups. The subgroup lattice index problem 863

Lemma 2.1. Let G be a group with K cD G and ½G=K finite cyclic. Then

Yn ½G=K¼ ½Ci=K; i¼1 where each ½Ci=K is a maximal chain. Also if K o Ai c Ci, then

Yn ai jG : Kj¼ jAi : Kj ; i¼1 where ai is the length of ½Ci=K.

Proof. Without loss of generality, we may assume that KG ¼ 1. Then by [8, Theorem 6.2.17]

G ¼ P1 Pt T; ð1Þ

K ¼ Q1 Qt ðT V KÞ;

ni1 and ¼ Si z Qi is a Pðni; piÞ-group, with Si elementary abelian of order pi , jQij¼qi, a prime of pi 1, and where T V K is a quasinormal subgroup of G (i.e. permutes with every subgroup). Also (1) is a Hall factorization, i.e. the factors have mutually coprime orders. Since ½G=K is cyclic, ni ¼ 1 for all i. Clearly ½T=ðT V KÞ G ½TK=K, a cyclic lattice. Now T V K is core-free in T and so T V K lies in the hypercentre Z of T, by [5]. Since T=Z must be cyclic, it follows that T is nilpotent. Then the lemma follows. r

We need a combinatorial result from finite set theory, which can be found in [2, (2.1.1)]: if X is a finite set that is the union of Xi with i a I (and I finite) then X T jJjþ1 jXj¼ ð1Þ j j a J Xjj: ð2Þ fAJ H I

This allows us to make a simple deduction from Lemma 2.1.

Lemma 2.2. Let K c DG with ½G=K finite. Let f½Hi=Kji a Ig be the maximal cyclic intervals in ½G=K. Then X T jJjþ1 jG : Kj¼ ð1Þ jð j a J HjÞ : Kj: fAJHI

Thus jG : Kj can be determined from the lengths of the maximal chains above K and the indices of K in the minimal non-trivial subgroups of these chains.

Proof. For g a G we have gK a fuKju a3g; K4g H Hi=K, for some i. Then the lemma follows from (2) and Lemma 2.1. r 864 S. Stonehewer and G. Zacher

We can now prove the first of our main results.

Proof of Theorem 1. We have a group G such that ½3g4=1 is a d.i. interval for all g a G, and we must prove that ‘ðGÞ is a d.i. lattice. Let Y c X c G with jX : Yj finite. We have to show that this index can be de- termined from ‘ðGÞ. Then we may assume that Y cD X. For, jX : YX j is finite and so there is a subgroup K cD X with K c Y and ½X=K finite. Thus jX : Kj is finite and jX : Yj¼jX : Kj=jY : Kj. By Lemma 2.2, it su‰ces to find jA : Yj for all A with Y o A c X.IfY p A, then A ¼ Y3g4 for some cyclic subgroup 3g4. But in any case, from Schmidt’s theorem (see (1)), there is always a cyclic subgroup 3g4 such that Y3g4 ¼ A.SojA : Yj¼j3g4 : 3g4 V Yj. However, for any cyclic subgroup 3x4 in A we have j3x4 : 3x4 V Yj¼j3x4Y : Yj c jA : Yj.So

jA : Yj¼maxfj3x4 : 3x4 V Yjjall x a Ag: r

The proof of Theorem 2 is considerably more di‰cult. We begin with a lattice- theoretic description of certain finite metacyclic groups. Recall that a subgroup lattice and the corresponding group are said to be modular if every element of the lattice, i.e. subgroup of the group, is Dedekind.

Lemma 2.3. Let G be a finite group. Then G is a product of a cyclic normal p-subgroup N and a cyclic q-subgroup K, where p and q are distinct primes, but G is not cyclic and not a P-group, if and only if the following conditions are satisfied: (i) ‘ðGÞ is irreducible and contains two maximal chains ½N=1 and ½K=1, not both of length 1, such that G ¼ 3N; K4, where N is a PðGÞ-invariant complement of K in G; and (ii) if ½N=1 has length d 2 and X, Y, K1, M are the subgroups of G defined by 1 o K1 c K,1o X o Y c N, and Y o 3Y; K14 ¼ M, then M is either cyclic or not modular with ½M=1 Z ½D8=1. (Here D8 is the of order 8.) Moreover, in this situation, the centre of G is the C in K such that ½N; C¼1, i.e. ½NC=1 is reducible; and ½N=1 is a d.i. interval.

Proof. For the necessity of the conditions, suppose that G ¼ N z K, where N is cyclic of order pa and K is cyclic of order q b,withp A q. Also G is not cyclic and not a P-group. Then certainly ‘ðGÞ is irreducible. It follows from Suzuki’s work on singular projectivities of finite groups (see [8, Theorem 4.2.6]) that G has no singular autoprojectivities. Thus N is PðGÞ-invariant and (i) follows. If ½N=1 has length at least 2 and M is not cyclic, then K1 acts non-trivially on Y and so M is not modular. Also ½M=1 Z ½D8=1, because D8 has no singularities (see [14, Theorem 12]). There- fore (ii) is true. Conversely suppose that conditions (i) and (ii) hold. Since N p G and N and K are not both , G is not a P-group. Suppose that G is a p-group. If N is an atom, then G ¼ N K, contradicting the hypothesis that N is PðGÞ-invariant. On the other hand, if ½N=1 has length at least 2, then M in (ii) has order p3, and this contradicts the hypothesis concerning M. The subgroup lattice index problem 865

Therefore G cannot be a p-group. Also, since ‘ðGÞ is irreducible, G cannot be cyclic. Thus we have established the su‰ciency of (i) and (ii). Finally suppose that these conditions are satisfied. Then clearly C ¼ ZðGÞ c K, C ¼ CK ðNÞ, and C is maximal in K such that ½NC=1 is reducible. Let L be such that C o L c K and let T ¼ NL. Then FðTÞ¼Np C and T=FðTÞ is a Pð2; pÞ- group. Therefore p is determined. r

Next we consider lattices with an infinite cyclic element which, together with each cyclic element, generates a .

Lemma 2.4. Let G be a group for which L ¼ ‘ðGÞ has an infinite cyclic element 3z4 such that 3x; z4 is a for all 3x4aL. Then the set pð3x; z4Þ of prime of the orders of the elements in the locally finite radical (i.e. the maximal nor- mal locally finite subgroup) of 3x; z4 depends only on the lattice ‘ð3x; z4Þ; and these primes can be determined. Moreover (i) for all 3x4aL, we can find all indices jH : Kj with K o H c 3x4 such that jH : Kj a pð3x; z4Þ; and (ii) if there is an infinite cyclic subgroup 3b4aL such that 3b4 V 3z4 ¼ 1, then ‘ðGÞ is a d.i. lattice.

Proof. The structure of 3x; z4 was described by Iwasawa (see [8, Theorem 2.4.11]). In particular the periodic elements form a normal abelian subgroup R and z acts on each p-component as a p-adic power. Let 1 o A c R. Then 3A; z4 is abelian and we can find K < 3z4 such that 3A; z4=K is primary. Thus jAj is the corresponding prime. So pð3x; z4Þ depends only on ‘ð3x; z4Þ. (i) If j3x4j is finite, then 3x4 c R and the argument above applies. Therefore we suppose that j3x4j is infinite and let K o H c 3x4. Then jH : Kj a pð3x; z4Þ if and only if there is an atom A in R such that 3H; A4=K is elementary abelian in Pð2; pÞ. (Recall that 3H; A4 is abelian.) So (i) follows. (ii) We have B ¼ 3b; z4 ¼ 3b4 3z4 G Cl Cl. Therefore for all primes p and n n d 1, we can find B p in ‘ð3b; z4Þ. Suppose that 3x4 V B A 1. Then given M o 3x4 V B, there is a unique prime p n such that M d 3x4 V B p for su‰ciently large n.Soj3x4 V B : Mj¼p and it follows that ½3x4=1 is a d.i. interval. Now suppose instead that 3x4 V B ¼ 1. If j3x4j is finite, then (i) applies; and if j3x4j is infinite, then we can take 3b4 ¼ 3x4 above. The result follows by Theorem 1. r

Clearly an atom of a non-abelian P-group is either normal or self-normalizing. Our next result concerns this situation.

Lemma 2.5. Let A be an atom of a group G and let

HA ¼fHi j A c Hi c G; Hi a Pðni; piÞ for some ni and pig: 866 S. Stonehewer and G. Zacher

If A p Hi for some i, then jAj¼minfpij all Hi a HAg. (Note that ni may be infinite here.)

Proof. For all i we have jAj¼pi or qi, where

G G Hi Cpi Cpi or Hi ðCpi Cpi Þz Cqi :

Since qi divides pi 1, we have jAj c pi for all i. But jAj¼pi for some i and the result follows. r

Remarks. (1) The class P consisting of finite groups of prime-power order and finite P-groups is lattice-defined; i.e. if ‘ðG1Þ G ‘ðG2Þ and G1 a P, then a P. This was proved in [13] (see also [8, Theorem 7.4.10]). (2) Let G a P and suppose that ‘ðGÞ is not a chain and G is not a P-group. If H c G, then jHj¼pm, where p is given by G=FðGÞ a Pðn; pÞ and m is the length of ½H=1. The proof is straightforward.

Our final result in this section uses the above lemmas and concerns infinite cyclic extensions of locally finite groups. It is a significant step on the way to proving The- orem 2.

Lemma 2.6. Let H be a normal locally finite subgroup of a group G with G=H infinite cyclic. Then ½3h4=1 is a d.i. interval for all h a H.

Proof. We may assume that 1 o 3h4 ¼ A, say. Also it follows from [11, Lemma 2.3] that for all s a PðGÞ, Hs ¼ H and

s is index-preserving on H: ð3Þ

We distinguish two cases.

Case 1. Suppose that there is a PðGÞ-invariant subgroup N of H such that N V A ¼ 1 and either (i) ‘ðH=NÞ a Pð; pÞ for some p, or (ii) ‘ðH=NÞ is locally cyclic and pri- mary. First suppose that (i) holds. We may assume that there is a maximal subgroup P=N of H=N such that P is PðGÞ-invariant. (Otherwise H=N is an elementary abelian p-group of rank at least 2 and jAj¼p.) Then H=P has prime order and is the locally finite radical of G=P. Let G ¼ 3H; x4. Certainly C ¼ C3x4ðH=PÞ can be recognised in ‘ðG=PÞ. Thus Lemma 2.4 (i) applied to ‘ðHC=PÞ gives jH=Pj. Therefore the order of every atom of H=N, and hence jAj, can be determined. Now suppose that (ii) holds. Then AN=N is PðGÞ-invariant. Let C ¼ C3x4ðAN=NÞ. Applying Lemma 2.4 (i) again, this time to ‘ðANC=NÞ, gives jAN=Nj¼jAj.

Case 2. Suppose that there is no subgroup N with the properties described in Case 1. Let F be any finite subgroup of H containing A and let S be a Sylow subgroup of F The subgroup lattice index problem 867 containing A.IfS is not elementary abelian or cyclic, then jAj is determined by Remark 2 above. (In this case S can be recognised in ‘ðFÞ.) Therefore we may as- sume that

S is elementary abelian or cyclic.

Suppose that NF ðSÞ¼CF ðSÞ for all F d A. Write jAj¼p. Then a well-known theorem of Burnside (see [7, (10.1.8)]) shows that F has a normal p-complement Fp0 . Thus [ N ¼ Fp0 F d A is a normal p0-subgroup of H, A V N ¼ 1, H=N is either elementary abelian or a locally cyclic p-group, and N is PðGÞ-invariant, by (3). But this contradicts the hy- pothesis of Case 2. Therefore we must have NF ðSÞ > CF ðSÞ, for some F d A. Then there is a non- trivial chain C such that S V C ¼ 1, S p 3S; C4 ¼ T, say, and ½T=1 is irreducible. First suppose that S is cyclic; thus T is metacyclic. If T is not a P-group, then ½S=1 is a d.i. interval, by Lemma 2.3, and so jAj is determined. On the other hand, if T is a P-group, then jAj is determined, by Lemma 2.5. Finally, if S is not cyclic, then again A is a normal subgroup of a P-group, and so again Lemma 2.5 applies. r

3 Further lemmas and proof of Theorem 2 We have considered infinite cyclic extensions of locally finite groups in Lemma 2.6. Now we look at a group G generated by a torsion-free modular subgroup A and one other element such that A cd G. In fact ‘modular’ here is equivalent to ‘abelian’, by Iwasawa’s theorem (see [8, Theorem 2.4.11]). We write

oðAÞ¼fp j p is prime and Ap ¼ Ag.

Lemma 3.1. Let G ¼ 3A; g4 be a group with A a non-trivial torsion-free modular sub- group, A cd G and A V 3g4 ¼ 1. (i) If jgj is finite, then ½3g4=1 is a d.i. interval. (ii) If jgj is infinite, then for all primes q c oðAÞ, we can recognise the subgroup M such that A c M o G and jG : Mj¼q.

Proof. As we stated above, A is abelian. (i) Clearly we may assume that g has prime order. Suppose that there is a modular subgroup H of G such that H > 3g4. Then H ¼ 3H V A; g4 and ½3g4=1 is a d.i. in- terval, by Lemma 2.4 (i). Therefore we may suppose that such an H does not exist. Then A p G, since otherwise 3A; Ag4 ¼ G and A V Ag would lie in ZðGÞ and would be non-trivial, so 3A V Ag; g4 would be abelian, a contradiction. 868 S. Stonehewer and G. Zacher

Choose a a A, a A 1, and let K ¼ 3a; g4 and B ¼ A V K. Since jK : Bj¼jgj, B is finitely generated (and torsion-free). Then there is a unique prime p such that K=B p2 is a finite p-group, not cyclic and not a P-group. By the Remarks after Lemma 2.5, p can be determined and jgj¼p. (ii) When jgj is infinite, A p G, by [10, Corollary 2.2]. Let q c oðAÞ.SoAq < A and A=Aq is elementary abelian. If the rank of A=Aq is 2 or more, then of course we see the prime q immediately; otherwise we do not. Let a a A n Aq, K ¼ 3a; g4 and B ¼ A V K. Denote factors modulo Aq by bars. Thus K ¼ KAq=Aq etc. We have K ¼ B z 3g4, a 2-generator metabelian group, with B an elementary abelian q-group and 3g4 infinite cyclic. By [3], K is residually finite. Therefore there is a subgroup N p K such that K=N is finite and 3a4 G N. Let L ¼ B V N.SoB=L is a finite ele- mentary abelian q-group. Let C be the centralizer of B=L in 3g4. Thus BC=L is the unique maximal modular subgroup of K=L containing B=L (using again Iwasawa’s structure of non-abelian modular groups with elements of infinite order; see [8, Theorem 2.4.11]). Therefore we can determine C. Thus we can also determine the maximal subgroup X of index q in C (from ‘ðBC=LÞ); and now we know q, even when A=Aq has rank 1. We have 3g4=X ¼ Q=X R=X;

0 o where Q=X is a q-group and R=X is a q -group. It follows that if Q1=X Q=X, then q Q1R is the subgroup of 3g4 of index q. Finally Q1R ¼ S=A , say, and AS ¼ M has index q in G. r

Corollary 3.2. If, in the notation of Lemma 3.1, joðAÞj c 1, then ½G=A G ½3g4=1 is a d.i. interval.

In the situation of Lemma 3.1 (ii) we can get more information if joðAÞj d 2, and more still if in addition rank(AÞ d 2.

Lemma 3.3. Let G ¼ 3A; g4 be a group with A ðA1Þ a torsion-free modular subgroup, A cd G, 3g4 G Cl and A V 3g4 ¼ 1. Suppose that joðAÞj d 2. Then (i) ½3g4=1 is a d.i. interval. Moreover, if rankðAÞ d 2, then (ii) ‘ðGÞ is a d.i. lattice.

Proof. (i) As in Lemma 3.1, A is abelian and A p G. Clearly we may assume that

i CAðg Þ¼1; for all i A 0; ð4Þ by Lemma 2.4 (ii). Let h be the map G ! PðGÞ defined by letting xh be conjugation in G by x for all x a G. Then h is a monomorphism of groups: ð5Þ The subgroup lattice index problem 869

For the of h is the set of elements in G which induce a power automorphism by conjugation. But since G is torsion-free and non-abelian, the identity map is the only power automorphism of G (see [8, Theorem 1.5.8]). Therefore (5) follows from (4). Write g ¼ gh. Let p a oðAÞ and define an automorphism a of A by

a: a N a p for all a a A:

Now extend a to an automorphism a~ of G by defining

a~: agi N a pgi for all a a A and i:

Then a~ fixes g and ja~j is infinite. Similarly let q a oðAÞnfpg, and obtain an auto- morphism b~ of G with q replacing p. Viewing a~ and b~ as elements of PðGÞ, at least one of 3a~4 and 3b~4 must be disjoint from 3g4. Suppose, without loss of generality, that it is 3a~4. Observe that g commutes with a~ and b~. Therefore we conclude, from the existence of h, that there is at least one projectivity s from G into PðGÞ such that

s 3g4 c Q G Cl Cl:

Thus, by Lemma 2.4 (ii), ½3g4s=1 is a d.i. interval. But s is index-preserving on ½3g4=1, by [11, Theorem 4.1]. Therefore ½3g4=1 is a d.i. interval and (i) is proved. (ii) We have rankðAÞ d 2. Let x a G, x A 1. If A V 3x4 ¼ 1, then ½3x4=1 is a d.i. interval, by (i). On the other hand, if A V 3x4 ¼ 3xi4A1, then Lemma 2.4 (ii) shows that ½3xi4=1 is a d.i. interval and hence so is ½3x4=1. Thus (ii) follows from Theorem 1. r

Suppose that A is a torsion-free modular subgroup of a group G.SoA is abelian. If A has rank 1 and ½G=A contains disjoint elements isomorphic to ‘ðClÞ, then we can now show that ‘ðGÞ is a d.i. lattice.

Lemma 3.4. Let G a D with hðGÞ d 3, and let A be a torsion-free modular subgroup ðhence abelianÞ of G. Suppose that A cd G and A has rank 1. Then ‘ðGÞ is a d.i. lattice.

Proof. By [10, Theorem A], A is quasinormal in G, and so A is an ascendant sub- group of G, by [9, Theorem A]. In fact A p AG, as is pointed out in the former refer- ence. By Theorem 1, it is su‰cient to show that ½3g4=1 is a d.i. interval, for all g a G. We distinguish three cases.

Case 1. Suppose that jgj is finite. Then Lemma 3.1 (i) proves our claim.

Case 2. Suppose that jgj is infinite and A V 3g4 ¼ 1. If joðAÞj c 1, then the result fol- lows from Corollary 3.2. On the other hand, if joðAÞj d 2, then it follows from Lemma 3.3 (i).

Case 3. Suppose that jgj is infinite and A V 3g4A1. Let 3a4 ¼ A V 3g4. Then it suf- fices to show that ½3a4=1 is a d.i. interval. Let N ¼ NGðAÞ. Since infinite cyclic sub- 870 S. Stonehewer and G. Zacher groups intersecting A trivially all normalize A (from [9, Lemma 2.1]), it follows that hðNÞ¼hðGÞ. Thus we may assume that A p G. We have A c CGðAÞ¼C, say, and G=C embeds in Q (the of non-zero rationals), since rank(AÞ¼1. Suppose that C=A is not periodic. Then there is an infinite cyclic subgroup 3x4 c C such that 3A; x4 ¼ A 3x4.So½3a4=1 is a d.i. interval, by Lemma 2.4 (ii). Therefore we may assume that

C=A is periodic.

Let Ci ¼ 3ci4, i ¼ 1; 2, be infinite cyclic subgroups of G, intersecting C trivially and disjoint modulo C. Then c1 and c2 act by conjugation on A. Let the action of ci be multiplication by mi=ni, i ¼ 1; 2 (using additive notation for A). a b Let p be any prime. Then there is an element c3 ¼ c1c2 , of infinite order, such that the action of c3 on A is multiplication by m3=n3, where p does not divide m3n3. Write

H ¼fu=v j u=v a A; p F vg:

(Here we are thinking of A as embedded in Q.) So H p 3A; c34, A=H is a locally cyclic p-group and \ i Hp ¼ 1: id1

pi Let fHl j l a Lg be the unique composition series of A refining fH ji d 0g. Each factor has order p and each Hl is normalized by c3. Now let 3b4 o 3a4. Then there is a unique prime p and the associated subgroup H such that a a Hlþ1nHl and b a Hl, for some l. Therefore

Hlþ1=Hl ¼ 3a4Hl=Hl G 3a4=ð3a4 V HlÞ¼3a4=3b4:

By Lemma 2.6 applied to the group 3Hlþ1; c34=Hl, the prime p is determined. It fol- lows that ½3a4=3b4 is a d.i. interval and hence so is ½3a4=1. r

Theorem 2 will now follow from our final result, via Theorem 1.

Lemma 3.5. Let G a D and suppose that hðGÞ d 3. Then for every element g a G, ½3g4=1 is a d.i. interval.

Proof. Let R be the locally finite radical of G.Ifg a R, then the result follows by Lemma 2.6. Therefore we may assume that g c R. Let A=R be a non-trivial torsion- free modular subgroup of G=R, with A cD G. (We know that A exists, by Proposi- tion 1.1.) Thus A=R is abelian. We claim that

½3g; R4=R is a d.i. interval: ð7Þ The subgroup lattice index problem 871

Then ½3g4=ð3g4 V RÞ is also a d.i. interval. But ½ð3g4 V RÞ=1 is a d.i. interval, from above, and hence so is ½3g4=1. Therefore it su‰ces to prove ð6Þ, and for this purpose we may assume that R ¼ 1. Also we may assume that

jgj is infinite, since otherwise Lemma 3.1 (i) applies. We distinguish three cases.

Case 1. Suppose that rankðAÞ¼1. Then the result follows from Lemma 3.4.

Therefore we may assume that

rankðAÞ d 2 and 3g4 V A ¼ 1, since otherwise Lemma 2.4 (ii) applies.

Case 2. Suppose that joðAÞj c 1. Then Corollary 3.2 shows that ½3g4=1 is a d.i. interval.

Case 3. Suppose that joðAÞj d 2. Here all hypotheses of Lemma 3.3 (ii) are satisfied and the result follows. r

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Received 8 July, 2008 Stewart Stonehewer, University of Warwick, Coventry CV4 7AL, England E-mail: [email protected] Giovanni Zacher, Universita` degli Studi di Padova, 35131 Padova, Italy E-mail: [email protected]