Modular permutations on <span class="mathjax-formula">$\mathbb {Z}$</span>

Modular Permutations on Z.

FRANCESCODELCASTILLO(*)

ABSTRACT- In this paper the groupMof permutationssofZfor which an integer nˆn(s)>0exists such that (z‡n)sˆzs‡nfor everyz2Zis studied.Mis countably infinite locally (abelian-by-finite) and contains all finitely generated (abelian-by-finite) groups as subgroups. The commutator subgroup M⁰is an infinite simple group and the quotient groupM=M⁰is isomorphic toZ. Finally, all abelian groups that can be represented as modular permutation groups are determined: these are countable abelian groups whose quotient over the torsion subgroup is free.

1. Definitions and first results.

Let s be a permutation of the set Z of the integers: we say that n2Nn0 is amoduleforsif

(z‡n)sˆzs‡n;

for everyz2Zor, equivalently, ifscommutes with the translationtnand in this case we say thatsismodular. If itexists, such anis notunique, since any multiple of it is still a module. The set of all modular permuta- tions is a group, denoted by M. For s2 M, we denote by mod (s) t he smallestmodule fors. Thennis a module forsif and only if mod (s) divides n. M_n is the subgroup of M whose elements are the permutations for which n is a module. A modular permutation s2Mn preserves the n- congruence relation, i.e. z1z2(modn) if and only if z1sz2s(modn).

Thus, a modular permutation s2M_n induces a permutations_n of then cosets ofZ=nZ. This fact allow us to write, for everyz2Z,

zsˆ(q_z;n‡a_[z]n)n‡([z]_n)s_n (1:1)

(*) Indirizzo dell'A.: Dipartimento di Matematica ``Ulisse Dini'', UniversitaÁ degli Studi di Firenze, Viale Morgagni, 67a, 50134 Firenze.

E-mail: [email protected]

whereqz;n, [z]nare resp. the quotient and the remainder of the (euclidean) division ofzbynanda[z]nis an integer depending only on [z]_n. With a slight abuse we adopt the same notation for both the remainder and the classes of n-congruence and will lets_n act on both these objects; for this reason we will choose the setf0;. . .;n 1gas support for the symmetric group S_nin place of the more usual f1;. . .;ng. Conversely we can associate to each r2Sn a permutation er2Mn justby setting a[z]n0 in the 1.1:

zerˆqz;nn‡([z]_n)r. Thus, for everyn1, two homomorphism are defined ln:Sn !Mn

r7!er

fn:Mn!Sn

s7!s_n

such that lnfnˆ1. fn is therefore surjective and ln determines an iso- morphic copyfS_nof S_ninM. LetD_nbe the kernel off_n: a permutationl2D_n can be written, according to 1.1, as

zlˆz‡a[z]nn:

The features of l(which fixes each n-congruence class and acts ``inside'' each of them as a translation bya_[z]n) suggest us to introduce a special class of modular permutations: forn1 andI f0;. . .;n 1gthe permutation l_n;Idefined, forz2Z, by

zln;Iˆ z‡n if zi (modn) for somei2I z otherwise

is called modular translation. In caseIˆ figwe justwritel_n;i andl_n;i is called anelementary modular translation(shortenedEMT). Itis clear that l_n;IˆQ

i2Il_n;i and mod (l_n;i)ˆn. It is pure routine to check that

D_nˆ hl_n;ijiˆ0;. . .;n 1iand thatD_nis a free abelian group of ranknfor

which thel_n;iform a basis.

Furthermore, Mn splits as the semidirect productDn^jSfn: if x2Mn, once takensˆxfnln2Sfnone has immediatelyxs ¹2Dnand thenxˆls, with l2Dn;s2Sfn uniquely determined. To this particular factorization we will refer (unless otherwise stated) in the following when we say

``xˆls2M_n''. We also observe that this factorization means that Mis locally (abelian-by-finite).

Sfnin its action by conjugation onDnpermutes the elements of the basis according to the natural action of Sn, namely one has (l_n;i)^s~ˆl_n;is; thus M_nis isomorphic to the permutational wreath productZwr_nS_n. This al- lows us to state the first of two theorems about groups that can be em- bedded inM, which is somehow a converse of the local (abelian-by-finite) structure ofM.

THEOREM1.1. Let G be a finitely generated (abelian-by-finite) group.

Then G can be embedded inM.

PROOF. LetA^>!G!!Hbe exact, withAabelian andHfinite.Ais the directproductTF of its torsion subgroupTand a free abelian group of rank, say,k. PutNˆF_Gthe core ofFinG;Nis still free abelian of rankk.

The indexjG:Njis finite and thenUˆG=Nis a finite group of order, say,n.

According to the Kaluznin-Krasner theorem ([2], theorem 6.2.8.), Gis iso- morphic to a subgroup ofN wr U'Z^kwr U. We now show how t he lat t er can be embedded inM. First we can embed the standard wreath product in the permutational wreath product (via the Cayley regular representation for finite groups)Z^k wr_U S_n; now,Z^kwr_U S_nˆ(Z^k)^njS_n9Z^knjS_kn, where, in the last embedding, Snis identified with the subgroup of Sknact ing on t hen blocks fik;. . .;ik‡(k 1)g of length k. The theorem is proved, since Z^knjS_kn'Zwr_knS_kn'M_kn. p

2. The commutator subgroup ofM.

In this section we show thatM⁰ is an infinite simple group (Theorem 2.8) and that the quotient groupM=M⁰is isomorphic toZ(theorem 2.5).

For this purpose it is useful to have another way of writing the elements of M in terms of certain basic elements. Let Gn be the subgroup of Mn

generated by the modular translations whose modules dividen, Gn ˆ hl_m;ijmdividesn; iˆ0;. . .;m 1i

and let G_n be the image ofG_n through the homomorphismf_n. Since the image of lm;i is easily checked to be the cycle 0… m . . . n m† of lengthn=min Sn, we haveGnˆ h…0 m . . . n m† jmdividesni.

THEOREM2.1. For every n2, if Gn ˆ(Gn)fn, the following equality holds:

G_nˆ S_n if n is even and not a prime power A_n if n is odd and not a prime power a p-Sylow subgroup of Sn if n is a prime power

8<

:

In order to prove the theorem we recall a result due to Jordan ([4], Theorem 13.9):

THEOREM(Jordan, 1873). Let p be a prime and G a primitive group of degree nˆp‡k withk3. If G contains an element of degree and order p, then G is either alternating or symmetric.

PROOF OFTHEOREM2.1. The firsttwo cases (nnota prime power) can be proved via Jordan's theorem. Letnˆpqkwithp<qdifferentprimes andk2N. Since the cycle 0… qk. . . n qk† 2Gnis an elementof order and degreep, one just has to check thatG_nis primitive. Letf0g 6ˆDbe a block containing 0 and kˆl_1;0f_nˆ…0 1 . . . n 1†. Let u be the smallestnon-zero elementofD; since 0k^uˆu,Dcontains the whole orbit of 0 under the action ofk^u and, since it is the smallest nonzero element, u mustbe a divisor ofn and we musthaveDˆ f0;u;. . .;n ug. Ifu6ˆ1, consider a divisorvofnsuch thatvis neither a multiple nor a divisor ofu.

If rˆl_v;0f_n, one has the contradiction 0rˆv62D, urˆu2D: then we musthaveuˆ1 andDˆ f0;1;. . .;n 1g, i.e.G_n is primitive. Now, ifnis odd the image of anylm;iis a cycle of odd length and therefore it lies in An; otherwise, ifnis even,Gncontains some odd permutation and must be the whole of S_n.

Proving the case ofna prime power is a matter of counting elements.

We recall that the order of a p-Sylow subgroup of S_pk is p^r, where rˆ1‡p‡ ‡p^{k 1}. We then proceed by induction on k2N. Let P_k ˆG_p^k. The case kˆ1 is trivial. Suppose now that P_k is a p-Sylow subgroup of S_p^kand considerP_k‡1S_p^k‡1. Denote again byP_k the image ofP_kvia the usual embedding of S_pkin the pointwise stabilizer of the set fp^k;p^k‡1;. . .;p^k‡1 1g in S_pk‡1, let kˆl_1;0f_pk‡1 and g2S_pk‡1 be a per- mutation such that, for 0zp^k 1, zgˆpz. LetP⁰_kˆ(Pk)^g; t hen

P⁰_k ˆ h((l_p^j_;0)f_p^k)^g j0j<ki ˆ h(l_p^j_;0)f_p^k‡1 j0<j<k‡1i andP⁰_khas supportf0;p;2p;. . .;p^k‡1 pg. For 0<ip 1, letPⁱ_kˆ(P⁰_k)^ki. ThePⁱ_k's are subgroups ofP_k‡1and have pairwise disjointsupports. Thus they generate their direct product Pˆ h(l_pj;i)f_pk‡1j0<j<k‡1; 0ip 1i, whose order isjP⁰_kj^pˆp^(p‡‡pk). Furt hermore,k62Pand (Pⁱ_k)^kp ˆPⁱ_k.

Letnow aˆ(lp;0)f_p^k‡1 and bˆak. We have 16ˆb2P_k‡1nP,b^pˆ1 and b normalizes P. Thus, hP;bi ˆP^jhbi and hP;bi has order p^(1‡p‡‡pk), i.e. itis a p-Sylow subgroup of S_pk‡1. Finally, since b acts on P essentially in the same way as k, we get P^jhbi ˆ hP⁰_k;ki ˆ

ˆ h(l_p^j_;0)f_p^k‡1j0j<k‡1i ˆPk‡1. p COROLLARY2.2. Ifnis even and not a prime power, thenGnˆMn. PROOF. Letx2Mn. Since the restriction offntoGn is surjective one can takeg2G_nsuch thatgf_nˆxf_n, and thenxˆgdfor somed2D_nG_n

and thereforex2G_n. p

COROLLARY2.3. Mis generated by the modular translations.

Thus, ifx2 M, besides the factorization xˆls in some Mn, we also havexˆQ

l^a_iⁱ with thel_i's elementary modular translations and thea_i's integers. The latter expression is not unique but will be useful in this section as well as in the next one.

Consider the application Cn:M !Z defined, for x2 M, by xCnˆⁿP¹

iˆ0(zx z).Cn is the sum of the shiftings of the first nintegers under the action ofx; the restrictionc_nofC_nt o M_nis a homomorphism, as a directcheck shows: letx;y2Mn, then

(xy)c_n ˆX^{n 1}

zˆ0

((zx)y z)ˆX^{n 1}

zˆ0

((zx)y zx)‡X^{n 1}

zˆ0

(zx z)ˆ

ˆX^{n 1}

zˆ0

[(a_[z]nn‡zx)y (a_[z]n‡zx)]‡xc_nˆX^{n 1}

zˆ0

(zy z)‡xc_nˆyc_n‡xc_n: Furthermore,Sfn ker(c_n) andln;ic_n ˆn. Thus ndividesxc_n for every x2Mn.

PROPOSITION 2.4. The application C defined, for x2 M, by xCˆ

ˆ lim

i!1

xCi

i is well defined and, ifx2Mn,xCˆxc_n

n . In particularCis a surjective homomorphism ofMontoZ.

PROOF. Letxbe an elementof M_nand writeiˆqn‡[i] (division ofi byn). We have

xCˆ lim

i!1

1 i

X

qn 1

zˆ0

(zx z)‡X^{i 1}

zˆqn

(zx z)

!

ˆ

ˆ lim

i!1

1

i q(xc_n)‡^{[i] 1}X

zˆ0

(zx z)

!

ˆ

ˆ lim

i!1

q(zc_n) qn‡[i]‡lim

i!1

P

[i] 1

zˆ0 (zx z)

i :

(2:1)

Ifzc_nˆknthe first limit of the last member of the (2.1) isk, while the second one equals 0, since ^{[i] 1}P

zˆ0 (zx z)

ⁿP¹

zˆ0jzx zj (an i-independent

constant). p

What can be observed in the last proposition is that, ifxˆQⁿ

iˆ1l^a_iⁱ is a factorization ofxin terms of EMT's, sincel_iCˆ1, thenxCˆPⁿ

iˆ1a_i. Thus, although the EMT's that occur in the factorization ofxare notuniquely determined, the sum of theai's is indeed unique. In a certain sense the EMT's play a role inMsimilar to that of transpositions in finite symmetric groups and one can also regard the value ofCatxas a sortof ``signature''.

However the usefulness of EMT's is clear in the following results.

THEOREM2.5. M⁰ˆkerCand thenM=M⁰'Z.

PROOF. The inclusionM⁰kerC is obvious sinceC has an abelian image. In order to prove the opposite inclusion, we note that, if xˆP^k

iˆ1l_i^ai2kerCwith thel_iEMT's, sinceP

a_iˆ0, we can suppose, up to multiplication by an element ofM⁰,a_iˆ( 1)ⁱandkeven (saykˆ2r).

Thus it suffices to show that, for every n;m2N and every possible i;j, l_n;i¹l_m;j2 M⁰.

Suppose then thatxˆl_n;i¹l_m;j. Ifnˆmˆ1 there is nothing to prove.

If nˆm2, let s2Sn be such that isˆj. Then one has l_n;i¹l_n;jˆl_n;i¹(l_n;i)^s~ˆ[l_n;i;es]2[Dn;Sfn]M⁰_n. We also note that [Dn;Sfn]ˆ

ˆkerC\Dn (since ford2Dn;es2Sfn, clearly (d ¹es ¹des)Cˆ0).

Finally, if n6ˆm, let sˆ2l:c:m:(n;m) and consider l_n;i andl_m;j as elements of Ms: bot hln;ifs andlm;jfs are odd permutation of Ss, and then l_n;i¹lm;j2AfsDs M⁰_sDs. Thusl_n;i¹lm;j2(M⁰_sDs)\kerCˆM⁰_s(Ds\kerC)ˆ

ˆM⁰_s[Ds;Ses]M⁰_s. p In order to obtain information about the commutator subgroupM⁰we need to better understand the structure of the groups Mn. From now on we assume n5 and setKn ˆkerC\Mn. Justby simple observations, as applying the isomorphism and the correspondence theorems, it is easy to verify the following facts:

Dn\Knˆ[Dn;Sfn]ˆ[Dn;Mn] is a free abelian group of rankn 1 with basisfl_n;0¹l_n;ijiˆ1;. . .;n 1g;

K_nˆ[D_n;M_n]^jSf_n; ifUnˆDn^jAfn, M⁰_nUn; M⁰_nˆ[D_n;M_n]^jAf_n^<₄K_n. LEMMA2.6. For n5,M⁰⁰_nˆM⁰_n.

PROOF. [Dn;Mn]M⁰⁰_n: it suffices to show that every generator l_n;0¹ln;i of [Dn;Mn] belongs to M⁰⁰_n. If 06ˆj6ˆi one can take s2An

such that jsˆj and 0sˆi. With this choices one has at once l_n;0¹l_n;iˆ[ln;0l_n;¹_j;es]2[[Dm;Mn];[fSn;Sfn]]M⁰⁰_n. Even more trivially Af_nˆ[Af_n;Af_n]M⁰⁰_n and putting things together gives M⁰_n ˆ

ˆ[Dn;Mn]^jAfnM⁰⁰_n. p LEMMA2.7. For n5, let H be a normal proper subgroup ofM⁰_n. Then H[D_n;M_n]. In particular H is abelian.

PROOF. SupposeH/M⁰_nandH6[D_n;M_n]. Since M⁰_n=[D_n;M_n] is sim- ple, we haveH[D_n;M_n]ˆM⁰_nand consequently

M⁰_n

H ˆH[Dn;Mn]

H ' [Dn;Mn] H\[D_n;M_n]:

Thus in particular M⁰_n=His abelian andHM⁰⁰_nˆM⁰_n. Thus,HˆM⁰_n. p The above results are enough to prove thatM⁰is simple.

THEOREM2.8. M⁰is a simple group.

PROOF. Let16ˆN be a normal subgroup of M⁰ and set, for every n5,NnˆN\M⁰_n.Nnis normal in M⁰_nand then must coincide with M⁰_n or be abelian and contained in [Dn;Mn]. We now show that the latter cannot always be the case. Suppose that, for everyn5,Nn[Dn;Mn].

Let m5 be a fixed integer and take xˆ^mQ¹

iˆ0 l^a_m;iⁱ 2N_m. If k>0, xis also contained in N_km and then one also has xˆ^kmQ¹

iˆ0 l^b_km;iⁱ . This means that, givenz2Z,x``fixes''zor ``moves'' itby a multiple ofkm, no matter how a large k>0 one chooses. This necessarily implies that xˆ1 and then Nˆ1, which contradicts the initial assumption N6ˆ1. Then for some m5 NmˆM⁰_m and, for every k>0, one has NkmˆM⁰_km, as NmN_kmimplies thatN_kmis notabelian. This conclude the proof since Nˆ S

k0N_kmˆ S

k0M⁰_kmˆ M⁰. 3. Abelian subgroups ofM.

Theorem 1.1 describes a large class of groups that can be embedded in M. In this section we focus our attention to abelian groups and will com-

pletely determine those abelian groups that can be represented as sub- groups ofM.

We denote by Fn the free abelian group of rankn( @₀). Given a se- quence (possibly infinite)Pof prime numbers,C_P is the group Dir

p2PC_pand C¹_P is the group Dir

p2PC_p¹.

Forg2 Mwe consider the equationx^kˆgand, if a solution exists, we call itak-rootofg. Itis clear thatmod (g) divides mod (x). Suppose now that xis ak-rootofgandgˆls;xˆdt2Mn; one hasgˆx^kˆd⁰t^k for some d⁰2Dnand in particulart^kˆs, i.e.tis ak-rootofs: in a certain sense this allows us to subordinate the existence of roots inM to the existence of roots in finite symmetric groups, for which the following result holds ([3]):

PROPOSITION3.1. s2Snhas ak-root if and only if for every positive integerl, the number ofl-cycles in the canonical decomposition ofsis a multiple offkg_l.

In the statement of the above proposition fkg_l denotes the part of k which shares prime divisors withl, i.e. ifPis the set of prime divisors ofl, fkg_lis theP-partofk.

If an elementgˆls2Mnadmits ak-rootx, itis notnecessary thatx lies in Mn: anyway, bothgandxwill be in some Mm, withma multiple ofn.

For this reason it is useful to know how to factorizegin the group Mmfor multiplesmofn.

LEMMA 3.2. Let n2N, i2 f0;1. . .;n 1g and k0. Then the fol- lowing equalities hold:

l_n;iˆl_{kn;i‡(k 1)n}(i i‡n . . . i‡(k 1)n);

l_n;i¹ˆl_kn;i¹(i i‡n . . . i‡(k 1)n) ¹:

Thus, if a is a positive integer,

l^a_n;iˆlkn;i‡(k 1)nlkn;i‡(k 2)n lkn;i‡(k a)n(i i‡n . . . i‡(k 1)n^a l_n;i^aˆl_kn;i‡n¹ l_kn;i‡2n¹ l_kn;i‡an¹ (i i‡n . . . i‡(k 1)n ^a:

For gˆls2M_n with lˆⁿQ¹

iˆ0l^a_n;iⁱ we put e(g)ˆⁿP¹

iˆ0ja_ij. We have the following technical result:

LEMMA3.3. Let gˆls2Mn, m2N. Thene(g^m)me(g).

PROOF. IflˆⁿQ¹

iˆ0l^a_n;iⁱ, one hasg^mˆll^s1l^sm‡1s^m andl^sj ˆⁿQ¹

iˆ0l^a_n;i^isj. Thus,

e(g^m)ˆX^{n 1}

iˆ0

X

m 1

jˆ0

a_is^j

X

i;j

ja_is^jj ˆmX^{n 1}

iˆ0

ja_ij ˆme(g): p

PROPOSITION 3.4. Let gˆls2M_n be an element of infinite order.

Then, for everyk >e(g), the equationx^k ˆgadmits no solution inM.

PROOF. As usual setlˆⁿQ¹

iˆ0l^a_n;iⁱ. We consider firstthe caseg2Dn, i.e.

sˆ1, and proceed by way of contradiction. Put aˆP

i ja_ij ˆe(g) and suppose that, for somek>a, an elementx2 Mexists such thatx^kˆl. We can regardxas an elementof some Mtnwithtlarge enough so thatt>ja_ij for every i and, if a_i6ˆ0, ka_i divides t. For such a choice of t we have gˆdt2M_tn, withtˆ ⁿQ¹

iˆ0t^a_iⁱ

and thet_i's are disjoint``t-cycles'' inSf_tn. Furthermore, when a_i6ˆ0, t^a_iⁱ is the product of ja_ij nontrivial cycles of lengtht=ja_ij(and in particulart6ˆ1). Letz_i(ˆz_t;i) be the number of cycles of lengtht=ja_ijin the decomposition oft: we havez_iⁿP¹

iˆ0ja_ij ˆafor every i, and this inequality holds for every choice oftwith the desired properties.

Now, ifxˆhy2M_tn, we havey^kˆt, i.e. tadmits ak-rootinSf_tnand, according to Proposition 3.1, forai6ˆ0,kˆ fkg_t=jaijdividesz_ia, which contradicts the initial assumptionk>a.

To conclude the proof, supposes6ˆ1 and letmbe the order ofs. Thus g^m2D_n. If x is a k-rootof g one also has x^kmˆg^m and then

kme(g^m)me(g) and thus,ke(g). p

As a consequence of the previous proposition we can easily describe the structure of a torsion-free abelian subgroup ofM.

THEOREM3.5. Let G be a torsion-free abelian subgroup ofM. Then G is free.

PROOF. By Proposition 3.4 everyg2Gadmitsk-roots only for a finite number of indexesk. SinceGis torsion-free the equationx^kˆgadmits at mostone solution inG. Then every elementofGadmits only a finite number of roots and this is enough to conclude thatGis free. p

Examples of free abelian subgroups ofMwe have found so far are the Dn's, of finite rankn. A natural question is to ask whetherMhas a free abelian subgroup of infinite rank and such a question can be answered positively. Such a subgroup can be defined as the union of an ascending chain of free abelian subgroups of M: letm2 be a fixed integer, set x0ˆ0 andG0ˆ hlm;0i; fori1 letxibe such that

0x_i2ⁱm 1

x_i6x_j (mod 2^jm); for everyj<i: (

If we defineG_iˆ hl₂jm;xj jjˆ0;. . .;ii, the groupGˆ S

i2NG_iis free abelian of infinite countable rank, i.e. G'F@0. Further, if one ``removes'' the generatorl_m;0from the generating set ofG, a groupHis generated which is still isomorphic to F_@0and whose supportis disjointfrom the setmZ.

We turn our attention to torsion subgroups ofM. We have already seen thatMcontains every finite group as a subgroup (Theorem 1.1) and a natural embedding one can think of is the regular representation in somefSn. If we adopt an argument similar to that we have used above for the free abelian groupGof infinite rank, we can easily embed inMdirectproducts of coun- tably many finite cyclic groups. LetP ˆ(p₁;p₂;. . .) be a sequence of prime numbers; for every primepiwe can find a modular permutation of orderpiin such a way that these permutations have pairwise disjoint supportsSi. Let n>p1and considerte12Sfnwheret1ˆ…0 1 . . . p 1† 2Sn. Fori2 consider:

a1;. . .;api the smallest positive integers not contained inⁱS¹

jˆ1Si; k_ithe smallest positive integer such that2^kin>a_pi;

t_iˆ…a₁ a₂ . . . a_pi† 2S₂kin.

Thentei2Sg₂kin is an elementofMof orderpiand thetei's generate a subgroup ofMisomorphic toCP. With small changes to the argument, as restricting the domain where to choose the integersa_i, one can constrain the support of the generated subgroup in any infinite subset of Zwith some modular feature, for example the set mZ mentioned above. It is therefore clear thatM contains some subgroup isomorphic to F@0CP, for every countable sequencePof prime numbers.

One can go further and consider a single g image (under the homo- morphisml_n) of a cycle of prime lengthpin the symmetric group S_n. As an elementingS_pn,gis the image of a permutationx2S_pnwhich is the product of pcycles of lengthp:xclearly admits ap-rootx₁in S_pnandx₁ is a cycle of

lengthp². Thus,gadmits ap-rootg1ˆxe1ingSpn. One can iterate the argu- mentand find, for everyi2N, an elementg_i2Sg_pin such thatg^p_i ˆg_{i 1}:g and thegi's generate a subgroup ofMisomorphic to the PruÈferp-groupCp¹. By applying this argument to each of thete_i's above, one can build a subgroup ofMisomorphic toC_P¹and, again, one can confine the support of the group into the setmZ. Since any abelian torsion group can be embedded in a direct productof PruÈferp-groups, one has immediately the following results.

PROPOSITION3.6. Mcontains every countable abelian torsion group as a subgroup.

PROPOSITION 3.7. Let P be a countable sequence of prime numbers andA_PˆF_@0C_P¹. ThenA_P is isomorphic to a subgroup ofM.

Proposition 3.7 individuates a large class of abelian groups that can be embedded in M: t he A_P's and their subgroups. We now show that all abelian subgroups of M belong to that class. Suppose A is an abelian subgroup ofM, letTbe its (maximal) torsion subgroup and consider the quotient groupA=T. Let16ˆaT2A=T(in particularais of infinite order inA) and supposexTis ak-rootofaTinA=T: for somet2T,x^ktˆa. If jtj ˆn, x^knˆaⁿ and, according to Theorem 3.4, kne(aⁿ)ne(a) and thenke(a). As a consequence, every elementofA=Tadmits only a finite number of roots and thenA=Tis free abelian.Tis therefore a direct factor of AandAis isomorphic to a subgroup of someAP as defined in Propo- sition 3.7. What we have seen so far proves the following theorem.

THEOREM3.8. Let A be a countable abelian group and T be its torsion subgroup: A can be represented as a subgroup ofMif and only if A=T is free abelian.

Since we have showed (theorem 2.8) that the commutator subgroupM⁰ of M is simple, a natural question could arise, whether it is possible to restate the last and theorem 1.1 withM⁰in place ofM.

For theorem 3.8 it is easily seen that it is just a matter of embedding F@0

inM⁰, since any embedding ofC_P¹already lies there. Such an embedding can be simply achieved by modifying the group Gˆ S

i2NG_i constructed before: one considers fG_iˆ h(l_2m;x1) ¹l₂^j_m;xj jjˆ2;. . .;ii M⁰_n and Geˆ S

i2

fG_i M⁰ which is still isomorphic to F_@0 with support disjoint from mZ.

For theorem 1.1 the same is not as immediate as for theorem 3.8, but still possible. The homomorphism f :Dn !D2n, defined by (ln;i)f ˆ

ˆl_2n;i(l_2n;i‡n) ¹, is injective and commutes with every automorphism of Sf_n; furthermore, (D_n)f [D_2n;Sf_2n]ˆD_2n\ker (C)ˆD_2n\ M⁰. Thus, itis possible to embed M_nˆD_n^jSf_n into [D_2n;Sf_2n]^jSf_2n M⁰ and conclude that every finitely generated abelian-by-finite group can be embedded in M⁰(this also shows that every finitely generated (abelian-by-finite) group can be embedded into the commutator subgroup of a finitely generated (abelian-by-finite) group).

FINAL REMARK. Just before this work was submitted, a paper by M. R.

Dixon, M. J. Evans and H. Smith appeared ([1]) in which the existence of a locally (abelian-by-finite) simple group that is not locally finite is estab- lished; the authors construct such a group by means of a direct limit in- volving wreath products of direct products of infinite cyclic groups by finite alternating groups, which are in some way basic ``bricks'' not too dissimilar to the ones of the groupMof modular permutations. However, our work was originally intended with the main purpose of obtaining a better un- derstanding of the group M, whose definition sounds quite natural and reasonable, even though almost no traces were found in the literature.

REFERENCES

[1] M. R. DIXON- M. J. EVANS- H. SMITH,Embedding groups in locally (soluble- by-finite) simple groups, Journal of Group Theory9(2006), pp. 383-395.

[2] M. I. KARGAPOLOV - JU. I. MERZLJAKOV, Fundamentals of the Theory of Groups, Graduate Texts in Mathematics, vol.62, Springer (1979).

[3] N. POUYANNE,On the number of permutations admitting an m-th root, The Electronic Journal of Combinatorics9(2002).

[4] H. WIELANDT, Permutationsgruppe, Math. Inst. Univ. TuÈbingen (1955).

[Translated in english: Finite Permutation Groups, Academic Press, New York (1964). Reprinted inMathematische Werke, Walter de Gruyter, Berlin, Vol.1(1994), pp. 119-198.]

Manoscritto pervenuto in redazione il 10 luglio 2006