The Birkhoff-Neumann Embedding of Relatively Free Groups

The Birkhoff-Neumann Embedding of Relatively Free Groups (*).

R. BRANDL(**) - G. CORSITANI(***) - L. SERENA(***) To Professor Guido Zappa on his 90th birthday

ABSTRACT- LetGrbe the relatively free group of rankrin the variety generated by a finite groupH:In this paper we determine the minimum positive integert(r) for whichGr is embedded in the direct product oft(r) copies of H;whenH is a minimal non-abelian group or a dihedral groupDn(nodd). Moreover the finite nilpotent groups of classcare characterized as those finite groups for whicht(r) is bounded from above by a polynomial of degreec:

1.Introduction.

LetHbe a finite group, and letV ˆVar(H) be the variety generated byH. Forr1, letGrˆFr(V) be the relatively free group inVwithrfree generators (see [12, p. 9]). By a result of G. Birkhoff [1, p. 441] and B.H.

Neumann [11, p. 519] (see also [12, Theorem 15. 71]), we have an embed- ding

G_r,!H^jHjr

ofG_ras a subgroup of the direct product ofjHj^rcopies ofH.

(*)Mathematics Subject Classification: Primary 20E10; secondary 20D99, 20E26.

(**) Indirizzo dell'A.: Mathematisches Institut der UniversitaÈtWuÈrzburg, Am Hubland 12, 97074 WuÈrzburg, Germany.

E-mail: [email protected]

(***) Indirizzo dell'A.: Largo F.lli Alinari n. 15, 50123 Firenze, Italy.

E-mail: [email protected]

(***) Indirizzo dell'A.: Dipartimento di Matematica e Applicazioni per l'Architet- tura, UniversitaÁ degli Studi di Firenze, piazza Ghiberti n. 27, 50122 Firenze, Italy.

E-mail: [email protected]

In general, much less directfactors of H will suffice to get such an embedding of Gr. For example, letHˆZp be a group of prime orderp.

ThenV ˆAp;where Ae denotes the variety of all abelian groups of ex- ponentdividinge:Moreover,G_rˆZ^r_pand we haveG_rH^r.

In this paper, we shall consider the functiont(r) defined as follows. For every positive integerr, lett(r) be the smallest positive integertsuch that the relatively free groupGrof rankrinVcan be embedded as a subgroup ofH^t, the direct product oftcopies ofH. By the aforementioned result of Birkhoff-Neumann, t(r) exists, and we have t(r) jHj^r. IfHˆZ_p for a primep, the above showst(r)ˆr.

Note that the function tdepends on the groupHwhich generates V, and itis nota function ofValone. Indeed, we can haveVar(H1)ˆVar(H2), but the corresponding functionst1;t2 differ (see the example below).

Let H be a finite abelian group of exponent e; and set VˆVar(H):

ThenVˆA_e andG_rˆF_r(V) is homocyclic of exponenteand rankr:So t(r)ˆe r

l(H)

wheree(x) denotes the minimum integerxandl(H) is the number of invariant factors ofHwhose order is equal to the exponent ofH(see [3, p. 245]).

In this note we determine the structure of the relatively free groups in Var(H) and the precise values of t(r) for the minimal non-abelian finite groupsHand the dihedral groupDn (nodd). In the case of the dihedral groupHˆDp;pprime andrˆ2;this was already done by B. Fine [4].

The following theorem was suggested by an analysis of the material mentioned above.

THEOREM. Let H be a finite group. Then H is nilpotent of classc if and only if t(r)is bounded from above by a polynomial of degreec in r:

The proof of this, as we will see in Section 5, can be obtained as a consequence of results of Higman [6].

2.The free groups in Var (H), withHa minimal non-abelianp-group.

In this chapter we study the varietyV generated by a finite minimal non-abelian p-group H;that is a finite non-abelian p-group Hsuch that every proper subgroup and every proper homomorphic image of H is abelian.

From [8, p. 309] we can deduce easily the following

PROPOSITION1. Let H be a minimal non-abelian p-group. Then one of the following holds:

i) HˆQ8is the quaternion group of order8:

ii) HˆHa  ha;bja^pa ˆb^pˆ1;a^bˆa^{1‡pa 1}ifor somea2:

iii) H is of exponent p and order p³:

PROPOSITION 2. Let H be a minimal non-abelian p-group and let V ˆVar(H):If exp(H)ˆp^a;then

VˆVar(x^pa;[x^p;y];[x;y;z]):

PROOF. LetWˆVar(x^pa;[x^p;y];[x;y;z]):ClearlyVW:Note that Wis nilpotent of finite exponent, so it is locally finite. If the above inclusion is proper, we can choose a finite groupGinWnVof minimal possible order.

By minimality of Gwe have thatG is critical. ThereforeG is a p-group generated by two elements with cyclic centre (see [12, Theorem 51.35]). By the laws of the varietyWwe have thatGis of class two,G⁰is of exponentp andG⁰Z(G):AlsoG^pZ(G):SoG⁰is of orderpandF(G)Z(G):Since Gis generated by two elements, we haveF(G)ˆZ(G) and the index ofZ(G) inGhas to bep²:Itfollows thatZ(G) has indexpin every maximal subgroup ofG:So every maximal subgroup ofGis abelian. Moreover, sinceZ(G) is cyclic andjG⁰j ˆp;also every proper homomorphic image ofGis abelian.

ThusGis minimal non-abelian of exponentp^gwithga:

Ifaˆ1; thenGis of exponentp:Proposition 1 impliesGˆH2V;a contradiction.

IfHˆH_a;one of the following holds:

a) GˆHg;with 2ga:

b) Gis of exponentp:

c) GˆQ₈:

For a) we shall prove that all the groupsGˆHg;with 2ga;belong to the variety generated by H_a: For this it is sufficient to show that H_{a 1}2Var(H_a) for a3: In fact H_{a 1} is isomorphic to a subgroup of Lˆ(H_a hci)=D; where c is of order p^a andDˆ ha^{pa2(p 1)}c ^{pa 2}i: For this note that the elementsxˆa c;yˆbwithaˆaD;bˆbD;cˆcDsa- tisfy the following relations:

x^pa1 ˆ(ac)^{pa 1}ˆ(a^{pa 2}c^pa2)^pˆ(a^pa1)^pˆa^pa ˆ1;

y^pˆb^pˆ1;

[x;y]ˆ[a c;b]ˆ[a;b]ˆa^pa1ˆa^pa2c^{pa 2}ˆx^pa2:

For b) we can repeat the above construction to show thatGis isomorphic to a subgroup of

Lˆ(H₂ hci)=D;

wherecis of orderp²andDˆ ha^pc^pi;andx;yas above.

For c) an analogous argumentshows thatGis isomorphic to a subgroup of Lˆ(H₂ hci)=D;where cis of order 4;Dˆ ha²c²iandxˆa; yˆb c:

ThusG2Var(H_a)ˆV;a contradiction.

Finally, letHˆQ8:IfGˆQ8;thenG2V;and we have a contradiction as above. OtherwiseGˆH2is the dihedral group of order 8:In this case we can repeat the above construction withLˆ(Q₈ hci)=D;wherecis of order 4;Dˆ ha²c²i; xˆacandyˆbc:So we arrive at the same contra-

dictionG2V: p

PROPOSITION3. Let H be a minimal non-abelian p-group of exponent p^a;say, andV ˆVar(H):Let GrˆFr(V)be the free group of rank r inV:

Then we have

a) Gr=G⁰_ris homocyclic of exponent p^a and rank r:

b) G_r⁰is elementary abelian of rank1

2r(r 1):

c) jGrj ˆp^{12r(r‡2a 1)}:

PROOF. LetGˆGr:Parta) follows from [12, Theorem 13.52]. For b), note thatG is of class two, and hence, in the case exp(H)ˆp^a;we have [x;y]^pˆ[x^p;y]ˆ1 for all x;y2G: So, G⁰ is elementary abelian. As d(G)ˆr;we have thatG⁰is of rank r

2 ˆ1

2r(r 1):Partc) now follows

from a) and b). p

THEOREM4. Let H be a minimal non-abelian p-group, letVˆVar(H);

and let GrˆFr(V)the free group of rank r2inV. a) If exp(H)ˆp^a;a1;then G_rˆ[N]L where

Nˆ hn₁;n₂;. . .;n_ri Z_p^aZ_p. . .Z_p

|‚‚‚‚‚‚‚‚‚‚‚{z‚‚‚‚‚‚‚‚‚‚‚}

r 1

; and L is isomorphic to the free group Fr 1(V)of rank r 1inV:

b) [N;L⁰]ˆ1:

c) If Lˆ hx₁;. . .;x_{r 1}iwith free generators x₁;. . .;x_{r 1},then the action of x_ion N is given by

n^x₁ⁱ ˆ n₁n_i‡1 (iˆ1;. . .;r 1)

n^x_jⁱ ˆ nj (iˆ1;. . .;r 1;2jr):

d) Let xrˆn1. Then the elements x1;. . .;xr are free generators of Gr:

PROOF. LetGˆ[N]LwithNandLas described in the statement of the theorem. We show thatG2V andd(G)r:ThenGis an epimorphic image ofF_r(V):On the other hand it follows from Proposition 3c) and from the definition of G that jF_r(V)j ˆ jGj: From this we then can conclude GFr(V):

We firstshow d(G)r:ClearlyGˆ hn1;. . .;nr;x1;. . .;xr 1i: The re- lations of Part c) yield:

[n₁;x₁]ˆn₂;[n₁;x₂]ˆn₃;. . .;[n₁;x_{r 1}]ˆn_r:

Son2;. . .;nr2 hn1;x1;x2;. . .;xr 1i;and we getGˆ hn1;x1;x2;. . .;xr 1i:

Now we check thatGsatisfies the identities ofV. First, we show thatG is of class two. Write automorphisms ofNasrr-matrices in the obvious way. Note thatLacts onN as the subgroup

Aˆ

1 a₂ . . . a_r

0 1 0 . . .

... ... .. . ...

0 0 . . . 1

0 BB

@

1 CC A

a₂;. . .;a_r2Z_p

ofAut(N):HenceAis abelian, and soL⁰centralizesN:SinceGˆ[N]L, we haveG⁰ˆ[N;L]L⁰. By the definition of the action ofLonN, we have that [N;L]C_G(L). SinceN is abelian and [N;L]N;we get[N;L]Z(G).

ThereforeG⁰Z(G) and soGis nilpotent of class two.

We now check the lawx^pa ˆ1:First, note that [N;L]ˆ hn2;. . .;nriis elementary abelian. As G is of class two, for all n2N; x2L we have (nx)^paˆn^pax^pa[x;n]^{pa(pa 1)=2}ˆ1.

It remains to show that [x^p;y]ˆ1 is a law inG:SinceGis nilpotent of class two, the law is equivalent to [x;y]^pˆ1:Therefore itis sufficientto show thatG⁰is elementary abelian. We haveG⁰ˆ[N;L]L⁰;where [N;L]

is elementary abelian by a previous observation and L⁰ is elementary abelian by Proposition 3b).

For d) observe that, sinceGis the free group of rankrinV andGis finite, then the generatorsx₁;. . .;x_{r 1};x_rfreely generateG: p

LEMMA5. Let H andVbe as in Theorem 4. Then we have a1) If exp(H)ˆp, then t(r)1

2r(r 1):

a₂) If exp(H)ˆp^a, witha2;then t(r)1

2r(r‡1):

PROOF. LetGˆGrandDˆH^t(r):By Proposition 3b), we have thatG⁰ is elementary abelian of orderp^{12r(r 1)}:AsD⁰Z^t(r)_p we have that

0rankG\D⁰

G⁰ t(r) 1

2r(r 1):

…1†

So in both cases

t(r)1

2r(r 1):

Suppose now that exp(H)ˆp^awitha2:Note that V1(G=G⁰)(G\D⁰)=G⁰:

AsG=G⁰Z^r_pa;this yields that (G\D⁰)=G⁰is of rank greater or equal tor.

Using (1), this implies t(r) 1

2r(r 1)rank(G\D⁰=G⁰)r;

so that

t(r)1

2r(r‡1): p LEMMA6. Let H be a minimal non-abelian p-group.

a1) If exp(H)ˆp, then t(2)ˆ1:

a2) If exp(H)ˆp^a,a2;then t(2)ˆ3:

PROOF. Let GˆF₂(V): If exp(H)ˆp, we have GˆH by Proposi- tion 3. Suppose exp(H)ˆp^a witha2:By Lemma 5 itsuffices to show thatG,!HHH:For this we observe that by Proposition 3a) we have G=G⁰Zp^aZp^a:SoG=G⁰,!HH:Now we consider a normal subgroup K ofGsuch thatG=KHandK\G⁰ˆ1:Indeed, using the notation of Theorem 4 we chooseKˆ hn²₁n₂;x²₁n₂iifHis the quaternion group, and we setKˆ hx^p₁;n₂¹n^p₁^{a 1}iotherwise. SinceG⁰ˆ hn₂iis of orderpandn₂62K, we haveK\G⁰ˆ1:HenceGˆG=(K\G⁰),!HHH: p

THEOREM7. Let H andVbe as in Theorem 4 and set GrˆFr(V):

a1) If exp(H)ˆp, then t(r)ˆ1

2r(r 1):

a₂) If exp(H)ˆp^a;a2;then t(r)ˆ1

2r(r‡1):

PROOF. LetGˆGr andx1;x2;. . .;xrbe free generators of Gr; as in Theorem 4d). Sety_i;jˆ[x_i;x_j] fori>j:Observe that, with the notation of

Theorem 4, we have

n1ˆxr

and n_iˆy_{r;i 1} (iˆ2;. . .;r):

Then the elements x_i (iˆ1;. . .;r) and the y_i;j satisfy the following conditions:

x^p_i ˆy^p_i;jˆ[x_i;y_l;k]ˆ1

where 2lr; 1kl 1;in the case of exp(H)ˆp; and x^p_i^aˆy^p_i;jˆ[x_i;y_l;k]ˆ1

where 2lr; 1kl 1;in the case of exp (H)ˆp^a;a2:

By Theorem 4, we have that G is a splitextension Gˆ[N]L where LˆG_{r 1} is the free group of rank r 1 in the free generators x1;x2;. . .;xr 1 and

Nˆ hxri hyr;1i :: hyr;r 1i:

Moreover we have thatL⁰ centralizesN.

By Lemma 6, the claim is true forrˆ2:So letr3:Firstconsider the case in whichHis of exponentp:

Forkˆ1;. . .;r 1 set

S_kˆ hL⁰;X_k;Y_ki;

where

Xkˆ fx1;. . .;xr 1g n fxkg; Ykˆ fyr;jj1j<rg n fyr;kg:

Observe thatS_kis normal inG. In fact

NC_G(L⁰); X_k^xi L⁰X_k(iˆ1;. . .;r 1); X_k^xr X_kY_k:

MoreoverG=Skis a non-abelian group of class 2 and exponentpon two generators. SoG=SkH:

Now \^{r 1}

kˆ1

Sk

\Nˆ^r\¹

kˆ1

(Sk\N) SincehL⁰;X_ki LandLnormalizeshY_kiwe have

\

r 1 kˆ1

(Sk\N)ˆ^r\¹

kˆ1

hYki ˆ1:

By induction, G=NL is embeddable in the directproductof (r 1)(r 2)

2 copies ofH. HenceGis embeddable in the direct product of (r 1)(r 2)

2 ‡(r 1)ˆr(r 1)

2 copies ofH.

For the other cases we can proceed in a similar way. IfHis a minimal non-abelianp-group of orderp^a‡1 and exponentp^a other than quaternion group, forkˆ1;. . .;r 1 we choose

Xk ˆ fx1;. . .;xk 1;xkp;xk‡1;. . .;xr 1g;

Y_kˆ fy_r;1;. . .;y_{r;k 1};y_r;k¹x^p_r^{a 1};y_r;k‡1;. . .;y_{r;r 1}g Moreover we set

Xrˆ fx2;. . .;xr 1;x^p_rg; Yrˆ fy_r;1¹x^p₁^a1;yr;2;. . .;yr;r 1g:

IfHis the quaternion group, forkˆ1;. . .;r 1 we put X_k ˆ fx₁;. . .;x_{k 1};x_k²x²_r;x_k‡1;. . .;x_{r 1}g;

Y_kˆ fy_r;1;. . .;y_{r;k 1};y_r;kx²_r;y_r;k‡1;. . .;y_{r;r 1}g and we set

Xrˆ fx1;. . .;xr 1;x²₁x²_rg; Yrˆ fyr;1x²₁;yr;2;. . .;yr;r 1g:

In both cases we obtain as above \^r

kˆ1

Sk

\Nˆ^r\¹

kˆ1

(Sk\N)\ hYri ˆ\^r

kˆ1

hYki ˆ1:

HenceGis embeddable in the direct product ofr(r 1)

2 ‡rˆr(r‡1)

copies ofH. 2 p

3.The free groups in the variety generated by a minimal non-nilpotent group.

In this section we will use the following well known results:

LEMMA 8. Let H be a minimal non-nilpotent group (that is a finite group H such that every proper subgroup and every proper homomorphic image of H is nilpotent). Then Hˆ[N]Q is a Frobenius group in which Q

is cyclic of prime order q and N is an irreducibleZpQ-module, for some prime p6ˆq;on which Q acts faithfully.

LEMMA 9. Let H be as in Lemma 8. We haveVˆVar(H)ˆA_pA_q:If G_rˆF_r(V)thenjG_rj ˆp^{1‡(r 1)qr}q^r:

PROOF. See [7, p. 175 f.] and [12, 21.13].

THEOREM10. Let Hˆ[N]Q be a Frobenius group as in the beginning of this section, and assume that N is elementary abelian of order pⁿ. Let G_r be the free group of rank r in Var(H). Then t(r)ˆ(r 1)(q^r 1)

n ‡er n

; whereer

n

denotes the smallest integerr n.

PROOF. Let tˆt(r): Clearly jG_rj mustdivide jHj^tˆp^ntq^t: As jG_rj ˆp^{1‡(r 1)qr}q^r, comparing exponents, the latter condition implies

1‡(r 1)q^rnt and rt:

…2†

Now

r1‡q‡ ‡q^{r 1}ˆq^r 1 q 1 q^r

q 1; but, by Fermat's Theorem,ndividesq 1, so

q^r q 1q^r

n: It turns out that, forr2 the following holds:

rq^r

n 1‡(r 1)q^r

n :

Therefore the two conditions (2) forr2 are equivalentto:

t1‡(r 1)q^r

n ˆ(r 1)(q^r 1)

n ‡r

n:

Now, since n divides q 1, n divides also (r 1)(q^r 1). Thus t and (r 1)(q^r 1)

n are integers, so

t(r 1)(q^r 1) n ‡er

n : By [2, p. 129], we haveG_rZ_p Z^r_p ¹wrZ^r_q

:Splitting off the center of

the wreath product, we arrive at a decomposition GrˆEW0; where EZ^r_p, and W0 by [13, Theorem 6.1] is directly indecomposable. Ob- viously,Eis isomorphic to a subgroup ofH^e(rn). So it suffices to show that W0is embeddable inH^l, wherelˆ(r 1)(q^r 1)

n . Indeed, in this case,Gr

is isomorphic to a subgroup ofH^l‡e(nr):Thatis t(r 1)(q^r 1)

n ‡er n

;

and, comparing with the above relation fort, we obtain the theorem.

NowW0ˆ[M]Q^r, whereMis elementary abelian of orderp^{(r 1)(qr 1)}. ThusMis decomposable intolˆ(r 1)(q^r 1)

n irreducibleQ^r-modulesMi

(1il).

Let K_i be the kernel of the action of Q^r on M_i, then jK_ij ˆq^{r 1}: Moreover the subgroup

R_iˆ M

j6ˆi

M_j K_j

is normal inW₀, and we haveW₀=R_iHfor all indicesi. We claim that Dˆ\^l

iˆ1

R_iˆ1;

consequentlyW₀is isomorphic to a subgroup ofH^l. For this we first show thatDis aq-group. Letx2Dbe ap-element. Thenx2R_ifor alli, and so x2L

j6ˆiM_j. But the intersection of these groups is trivial, hencexˆ1:SoD is a normalq-subgroup ofW₀and, sinceW₀is indecomposable, we arrive at

Dˆ1 as claimed. p

4.The free groups in Var(D_n);nodd.

LetG_rˆF_r(Var(D_n)) be the relatively free group of rankr1 in t he variety generated by the dihedral group Dn where n is odd. Then Var(Dn)ˆAnA2 (see [9]). We haveGrˆ[N]Qwhere, by the theorem of Nielsen-Schreier, a minimal setof generators of N consists of r⁰ˆ

ˆ(r 1)2^r‡1 elements andQis an elementary abelian group of rankr:

By [12, 21.13] we have that jG_rj ˆ2^rn^{(r 1)2r‡1} and in particular N is homocyclic.

LEMMA11. Let GrˆFr(Var(Dn)):Then N is the direct product of r Q- modules each of which is homocyclic, r 1are regular modules (i.e. Q acts regularly on a minimal set of generators) and one is a trivial Q-module.

PROOF. By the preliminaries we have thatNis homocyclic of ordern^r0 (using the above notation). Letfp₁;. . .;p_sgbe the set of all distinct prime divisors ofnwithnˆp^a₁¹p^a₂² p^a_s^sandP_i2Syl_pi(N) where 1is:Then Qacts on eachP_i. letP_iˆP_i=F(P_i). By [5] P_i is the direct sum ofr 1 regularF_piQ-modules and a trivial F_piQ-module. LetP_i;j a regular direct factor ofP_iwith 1jr 1 andP_i;r the trivial direct factor. Denote by fv^i;j₁ ;. . .;v^i;j₂rga basis ofP_i;jon whichQacts regularly, such thatv^i;j_r ˆv^i;j₁a_r with a_r2Q; 1r2^r anda₁ˆ1:Letv^i;j₁ 2P_iwithv^i;j₁F(P_i)ˆv₁^i;j and v^i;j_r ˆv^i;j₁ a_r with 1i2^r: Moreover let P_i;jˆ hv^i;j_r j1r2^ri: Then

P_i;j=F(P_i;j)ˆP_i;j and P_i;jˆ hv^i;j₁i . . . hv^i;j₂ri. For,

P_i;jF(P_i)=F(P_i)P_i;j=(P_i;j\F(P_i)) and F(P_i;j)P_i;j\F(P_i): Then P_i;j P_i;j=F(P_i;j)

(P_i;j\F(P_i))=F(P_i;j)andP_i;j=F(P_i;j) cannothave dimension less than 2^rwhich is the dimension ofPi;j;Itfollows thatPi;j ˆ P_i;j

F(P_i;j):On the other hand, sincefv^i;j_rj1r2^rgis a basis forPi;j;we have thatfv^i;jr j1r2^rg is a minimal setof generators ofP_i;j:MoreoverP_iˆP_i;1P_i;2. . .P_i;r:Then jP_ij ˆ jP_i;1P_i;2. . .P_i;rj ˆ(p^a_iⁱ)^{2r(r 1)‡1}: Since jP_i;jj (p^a_iⁱ)^2r we musthave Pi;l\Pi;mˆ1 for all indices 1l<mr and jPijj ˆ(p^a_iⁱ)^2r for 1jr 1:In particular Pi;j is homocyclic for all jwith 1jrand

P_iˆP_i;1. . .P_{i;(r 1)}P_i;r: Let c_i be an elementof P_i such that

c_iˆc_iF(P_i)=F(P_i) is a generator ofP_i;rwhich is centralized byQ:ThenP_i;r is fixed byQ:The elements

fv^1;j₁ (v^2;j_r ) v^s;j_r j1jr 1 and 1r2^rg

and the elementc1 csare generators ofN;andQacts regularly on the set of elements

v^1;j₁ v^s;j₁ ;. . .;v^1;j₂r v^s;j₂r; 1jr 1: p

LEMMA 12. Let Grˆ[N]Q the relatively free group of rank r in the variety Var(Dn)with n odd. Then there is a baseB ˆ fv1;v2;. . .;vr⁰gof N such that every element of Q fixesr⁰‡1

2 elements ofBand inverts the other r⁰ 1

2 elements.

PROOF. By the previous lemma, we can restrict attention to a moduleN

on whichQacts regularly. We havejQj ˆ2^r:We proceed by induction onr:

First of all we see that the result is true forrˆ1:For, letB ˆ fv1;v2g a base of N on which Qˆ hai (jQj ˆ2) acts regularly. So v1aˆv2 and v₂aˆv₁:Consider the elementsv₁v₂ andv₁v₂¹:Thenhv₁v₂;v₁v₂¹i ˆN.

We have (v₁v₂)aˆv₂v₁and (v₁v₂¹)aˆv₂v₁¹ˆ(v₁v₂¹) ¹. Now letQbe an elementary abelian group of order 2^r(r>1) which acts regularly on a base Bof a homocyclic groupN:LetB ˆ fv1;. . .;v2^rg:We can assumeviˆv1ai

witha_i2Qanda1ˆ1:Moreover, we can order the elements ofQso that the set of elements fa₁;. . .;a₂^{r 1}g is a subgroup Q₀ of Q: Then [N]QN₀wr_Q0Q;is the twisted wreath product of the groupN₀generated by the elementsv₁;. . .;v₂r 1by the groupQ:By induction, we have thatN₀ possesses a baseB0ˆw1;. . .;w₂^{r 1}such that every element ofQ0fixes half of the elements ofB0and inverts all the others. Now letf1;bgbe a trans- versal ofQ₀inQ:We can identify a base ofNwith the set of ordered pair (w_i;d) where d can be either 1 or b. Consider the set C of elements f(w_i;1)(w_i;b);(w_i;1)(w_i;b) ¹gwith 1i2^{r 1}. Then Cis a base ofN:If a2Q0 then (wi;1)(wi;b)^eaˆ(wia;1)(wia;b)^e whereeis either 1 or 1:If a2QnQ0thenaˆhbwithh2Q0. Then (w_i;1)(w_i;b)^ehbˆ(w_ih;b)(w_ih;1)^e: So any elementofQfixes half of the elements ofBand inverts the others.p THEOREM13. Let HˆD_n with n odd and let G_r be the free group of rank r in Var(H). Then t(r)ˆ(r 1)2^r‡1:

PROOF. Let tˆt(r) and r⁰ˆ(r 1)2^r‡1: Clearly jGrj mustdivide jHj^tˆ2^tn^t:AsjGrj ˆn^r02^randr<r⁰we have, comparing exponentsr⁰t:

By Lemma 12 we haveG_rˆ[N]Q[N]Q^ where Q^ Z|‚‚‚‚‚‚‚‚‚‚{z‚‚‚‚‚‚‚‚‚‚}₂. . .Z₂

r⁰

acts onNas the group of all diagonal matrices with entries1:Itis easy to see that [N]Q^ (H)^r0:Sotr⁰:Hencetˆr⁰ˆ(r 1)2^r‡1: p

5.On the growth of the functiont(r).

We end this note with the proof of the theorem stated in the introduction.

PROOF OF THETHEOREM(¹) Let G_r be the free group in the variety

(¹) We thank an unknown referee for greatly improving upon our original result.

Var(H), and letjGrj ˆp^e₁¹ p^e_s^s be the prime decomposition of its order.

Assume that we have an embeddingh:G,!H^t(r):Letpi:H^t(r)!Hbe the i-th projection. Setf_iˆp_ih;and letK_i:ˆKer(f_i):Ashis an embedding, the intersection of all theK_iis the trivial group. By Lagrange's Theorem, already the intersection of at moste₁‡. . .‡e_sof these kernels is trivial.

We have

f(r):ˆlog₂jG_rj ˆlog₂p^e₁¹ p^e_s^s e₁‡. . .‡e_s; and hence we gett(r)f(r):

On the other handjGrj jHj^t(r);sof(r)t(r)log₂jHj:Itfollows thatif one off(r) andt(r) is bounded by a polynomial of degreecinr;then so is the other. By Higman's result [6, p. 154] we have thatHis nilpotent of class cif and only iff(r) is a polynomial of degreec: p

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Manoscritto pervenuto in redazione il 3 ottobre 2005.