( ∗ )-groups and pseudo-bad groups

Volumen 37 (2003), p´aginas 51–63

( ∗ )-groups and pseudo-bad groups

Luis Jaime Corredor Universidad de los Andes, Bogot´a

Abstract. We give an example of an infinite simple Frobenius group G without involutions, with a trivial kernel and a nilpotent complement. Nevertheless, this group is notω- stable (not even superstable), this is the ”only” property missing in order to be a counterexample to the Cherlin-Zil’ber Conjecture which says that simpleω- stable groups are algebraic groups.

Key words and phrases. Frobenius groups, group of finite Morley rank, pseudo- bad group, HNN-extension.

2000 Mathematics Subject Classification. Primary: 20A15. Secondary: 03C45, 03C60.

1. Introduction

In [Co2] we define a bad group to be a connected non-solvable group of finite Morley-rank in which all the definable proper subgroups are nilpotent-by-finite (i.e. all definable proper connected subgroups are nilpotent). We proved there that a bad group has a definable quotient which is a simple bad group. The Cherlin-Zil’ber conjecture states that every simple group of finite Morley rank is an algebraic group over an algebraically closed field, and we know that the max- imal connected solvable subgroups (called Borel subgroups) of a non-solvable algebraic group are non-nilpotent. Therefore, the existence of bad groups would be in contradiction with the conjecture above.

The structure of a simple bad group is well understood (see [Co2]): if G is such a group, it has a proper definable subgroup B, which is selfnormalizing, such that the union of all its conjugates is G and the intersection of any pair of them is trivial. B is in fact a Borel subgroup of G, it is nilpotent and we can

51

also prove that G does not have involutions. We call such a pair (G,B), where G is not necessarily of finite Morley rank, a pseudo-bad group.

Definition. Apseudo-bad groupis an infinite simple group G, without involu- tions, with a nilpotent proper subgroup B which is selfnormalizing, such that the union of its conjugates is G and the intersection of any pair of them is trikvial.

Equivalently a pseudo-bad group is an infinite simple Frobenius group G, without involutions, with a nilpotent complement B and with trivial kernel. It was unknown whether such a group existed.

In this article we construct a pseudo-bad group G, or rather a family of them, where the complement B is an infinite cyclic and definable subgroup and the maximal solvable subgroups of G are B and their conjugates. SinceB ∼=Z is definable, G is not of finite Morley rank ( orω−stable). We show that G is not even superstable.

S.V. Ivanov gave a similar example, but with the following additional prop- erty: there is anm (in his example m= 10⁶) such that, for everyb /∈ B and every conjugacy class C 6= {1} we have (bB)^m =C^m = G. This property is also satisfied by any simple bad group. For this reason his example is better than ours. His result was announced as an abstract in [I] but we do not know the actual proof of it. We presented our results for the first time in 1989 (see [Co1]) and we obtained them independently. We believe that our construction is simpler than Ivanov’s and therefore we present it here.

2.

-groups

Definition. A (∗)-groupis a torsionfree group G with the following property:

(∗) For all nontrivial element g∈G, CG(g) is cyclic.

In the rest of the paper, if G is a (∗)-group and g ∈Gr{1}, we will call vg

one of the generators of the cyclic groupCG(g).

Examples of (∗)-groups:

(1) hZ,+iis the only non trivial abelian (∗)-group.

(2) Free groups are (∗)-groups; cf [L-Sch].

(3) In Theorem 3 bellow we show that some HNN-extensions of (∗)-groups are also (∗)-groups.

(4) Subgroups of (∗)-groups are (∗)-groups.

(5) Free products of (∗)-groups are (∗)-groups. See Theorem 2 bellow.

Theorem 1. Let G be a(∗)-group,g∈G− {1}andvg∈Gso thatCG(g) = hvgi. Then:

(1) CG(vg) =hvgi

(2) CG(g) =hvgiis a maximal cyclic subgroup of G.

(3) For everyr∈Z^∗,CG(g^r) =CG(g).

(4) CG

¡CG(g)¢

=NG

¡CG(g)¢

=NG

¡hgi¢

=CG(g).

(5) For everyh∈Gr{1}: ifCG(h)∩CG(g)6=h1i, thenCG(h) =CG(g), and if ³ S

x∈G

CG(h)^x

´

∩ ³ S

x∈G

CG(g)^x

´

6= h1i, then S

x∈G

CG(h)^x = S

x∈G

CG(g)^x.

(6) The relationa↔b, “acommutes withb”, is an equivalence relation in Gr{1}.

(7) Ifu⁻¹g^pu=g^q, wherep∈Z^∗,q∈Zandu∈G, thenp=q.

(8) Letx, h∈Gsuch thatCG(x) =hxiandh^m=x^p, form, p∈Z,m6= 0.

Then m divides p andh=x^p/m. (9) Every solvable subgroup of G is cyclic.

Proof.

(1) There is ap∈Zsuch thatg=v^p_g, becauseg∈CG(g) =hvgi. Therefore CG(vg)⊆CG(g) =hvgi.

The other inclusion is clear.

(2) hvgiis a maximal cyclic subgroup of G: ifhvgi ≤ hui, thenucommutes withvg, so

hui ≤CG(vg)≤ hvgi ≤ hui.

(3) It is clear thatCG(g)≤CG(g^r). ButCG(g^r) is also a cyclic subgroup of G. By the maximality ofCG(g) , we have the equality.

(4) The following inclusions are clear CG(g)≤CG

¡CG(g)¢

≤NG

¡CG(g)¢

≤NG

¡hgi¢ . We show that NG

¡hgi¢

≤ CG(g): Let u ∈ NG

¡hgi¢

, i.e., g^u = g^±1. Then u² ∈ CG(g) i.e., u² =v^q_g for some q ∈Z. If u6= 1, i.e., q 6= 0, then we getCG(u²) =CG(u), so

u∈CG(u) =CG(u²) =CG(v_g^q) =CG(vg) =hvgi=CG(g).

(5) Let b 6= 1 in CG(h)∩CG(g). By (2) CG(g) and CG(h) are maximal cyclic subgroups of G. Both are contained inCG(b), which is also cyclic.

Hence

CG(h) =CG(b) =CG(g).

The second part of the claim follows from here.

(6) The relation “↔” is clearly reflexive and symmetric. The transitivity follows easily from (5).

(7) By hypothesis CG(u⁻¹gu)∩CG(g) 6= 1, soCG(g)^u =CG(g). By (4) u∈NG

¡CG(g)¢

=CG(g).Hence p=q.

(8) We have

h∈CG(h^m) =CG(x^p) =CG(x) =hxi.

Thenh=x^q, for someq∈Zandx^p=h^m=x^qm. Thereforep=qm, q=p/mandh=x^p/m.

(9) LetH 6= 1 be a solvable subgroup of G. H contains a non trivial abelian normal subgroupH⁽ⁿ⁾. Let 16=h∈H⁽ⁿ⁾, thenH⁽ⁿ⁾≤CG(h). Since CG(h) is cyclic, thenH⁽ⁿ⁾is also cyclic. LetH⁽ⁿ⁾=hui. Then

H ≤NG

¡hui¢

=CG(u), and H is cyclic. ¤X

Definition. Let G be a torsion-free group. x∈Gr{1}is called indecompos- ableifhxiis a maximal cyclic subgroup of G.

Let G be a (∗)-group and x∈Gr{1}. The set C_x^G:= [

g∈G

CG(x)^g

is calledthe component of x in G.xis called atomicifCG(x) =hxi.

Theorem 2. Let G and H be(∗)-groups. Then G∗H, the free product ofG andH, is also a(∗)-group.

Proof. Everyw∈G∗H,w6= 1, can be written uniquely as a productw1· · ·wn, wherew16= 1, eachwibelongs G or H and consecutive factorswi andwi+1are not in the same group. This is called the normal formof w and we say that w=w1· · ·wn isreduced. The number|w|=nis called the length ofw.

We callw=w1· · ·wn cyclicly reduced ifwn andw1 are in different factors orn= 1.

Let w ∈ G∗H, w 6= 1. The proof will be complete once we prove by induction on|w|=nthe following claim:

Claim. For everyr∈Z^∗,CG∗H(w^r) =CG∗H(w)is cyclic.

If |w| = 1. then w ∈ G or w ∈ H and so CG∗H(w^r) equals CG(w^r) or CH(w^r). Now we can apply Theorem 1 (3).

Suppose that|w| ≥2. Without lost of generality, assume that wis cyclicly reduced (every element ofG∗H is conjugated to one element which is cyclicly

reduced). We may also assume, without lost of generality, that wis indecom- posable; otherwisew=v^rfor somev∈G∗H,v indecomposable, also cyclicly reduced and |v| < |w|. But, if the claim holds for v, it holds forv^r = w as well. Therefore w = w1· · ·wm is cyclicly reduced and indecomposable. Let u=u1· · ·unbe inCG∗H(w^r). We are done if we prove by induction on|u|=n that

u∈ hwi¡

≤CG∗H(w)≤CG∗H(w^r)¢ .

First we show that either u = 1 or n ≥ m. Suppose, for a contradiction, that u 6= 1 and n < m. It follow by induction that CG∗H(u) is cyclic. Let CG∗H(u) =hvui.Thenu=v_u^q for someq∈Z^∗ and|vu| ≤ |u|. Therefore

CG∗H(vu) =CG∗H(v_u^q) =CG∗H(u) =hvui.

By hypothesis we have w^r ∈ CG∗H(u) = hvui. Then w^r =v_u^±s where s ≥ 0.

Hence

w∈CG∗H(w^r) =CG∗H(v_u^±s) =CG∗H(vu) =hvui.

Then w =v_u^p for some p∈ Z^∗. But w is indecomposable, so |p| = 1 and we have|w|=|vu| ≤ |u|<|w|, a contradiction.

Assume now that u 6= 1. We have n ≥ m and since u∈ CG∗H(w^r), the following equality holds:

u1· · ·un−m· · ·unw1· · ·wm· · ·w1· · ·wmu⁻¹_n · · ·u⁻¹₁ =w^r.

The word on the left side must be reducible for the words to have the same lenght. This can only happen ifunw1 or wmu⁻¹_n is reducible. It is clear that only one of them is reducible because w is cyclicly reduced. Therefore there are two cases, but we will consider only the case in whichwmu⁻¹_n is reducible.

Let i ≤ m−1 be maximal such that wm−i· · ·wmu⁻¹_n · · ·u⁻¹_n−i =: bi is an element of G or H. One can easily see thati=m−1; otherwisewm−i−1biu⁻¹_n−i−1 would be reduced and so would be the word

uw^r−1w1· · ·wm−i−1biu⁻¹_n−i−1· · ·u⁻¹₁ =uw^ru⁻¹=w^r. By a length argument we getm=|w|>|u|=n, a contradiction.

Let ˆu = u1· · ·un−mb⁻¹_m−1. Then u = ˆuw and so w^r = ˆuw^ruˆ⁻¹. Since

|ˆu|<|u|, by induction we get that ˆu∈ hwi. Thereforeu∈ hwi. ¤X

Now we make a brief introduction to HNN-extensions. Let G be a group with two isomorphic subgroups A and B. Letϕ:A−→B be an isomorphism.

TheHNN-extensionof G with respect to A, B andϕis the group G^∗=­

G, t; t⁻¹at=ϕ(a), a∈A® .

Each elementw∈G^∗can be written in the formw=w0t^²1w1· · ·t^²nwn, where thewi’s are in G and²i=±1. This representation ofwis called reduced if it does not contain the substringt⁻¹witwithwi∈Aor twit⁻¹ withwi∈B.

If we choose a system of right coset representatives for A and B in G, then we get for each element w ∈ G^∗ a normal form w = w0t^²1· · ·t^²nwn (n≥ 0) with the properties:

(a) w0is an element of G.

(b) If²i=−1, thenwi is one of the representatives for the cosets of A in G.

(c) If²i = 1, thenwi is one of the representatives for the cosets of B in G.

(d) t^²1t^−² is not a substring.

|w|=ndenotes the length of w. w=w0t^²1· · ·t^²nwn (n≥0) is calledcyclicly reducedif every cyclic permutation ofwis reduced.

Theorem 3.

(i) Let G be a(∗)-group,x,z∈Gr{1}be atomic elements with different components. Then the HNN-extensionG^∗ =­

G, t; t⁻¹xt=z® is also a(∗)-group and for everyu∈Gr{1}, CG^∗(u) =CG(u).

(ii) If y ∈ G is a G-component other than that of x or that of z, i.e.

C_y^G6=C_x^G andC_y^G6=C_z^G, thenC_y^G∗6=C_x^G∗ andC_y^G∗6=C_z^G∗.

Proof. We use the following conventions: y1 := x and y−1 := z. With this notation we have that t^−δy^m_δ t^δ =y_−δ^m, where δ =±1 and m ∈Z. If w ∈G,

²=±1 andδ=±1, thent^²wt^δ is reducible if and only if²=−δandw=y_δ^m for somem∈Z. We use the standard theory of HNN-extensions which can be found in, say, [L-Sch].

(i) ThatG^∗ is torsionfree follows from the general theory of HNN-extensions.

We show thatG^∗ has also the property (∗). We shall be done when we prove by induction on|w|the following claim:

Claim A. For every w ∈ G^∗r{1}, CG^∗(w) is cyclic (equal to hvwi) and CG^∗(w^r) =CG^∗(w)for everyr∈N^∗.

We consider two cases:

Case I.Supposew∈Gr{1}. It is enough to proof thatNG^∗

¡hwi¢

=NG

¡hwi¢ , because

NG

¡hwi¢

=CG(w)⊆CG^∗(w)⊆NG^∗

¡hwi¢ . Let u=u0t^δ1· · ·t^δnun ∈ NG^∗

¡hwi¢

be reduced in normal form. We want to show that n = 0. Suppose that n ≥ 1. We can suppose that un = 1 and w=y_−δ^p

n for some p∈Zsinceuwu⁻¹=w^r forr∈Z, i.e., u0t^δ1· · ·t^δnunwu⁻¹_n t^−δn· · ·t^−δ1u⁻¹₀ w^−r= 1

which implies that t^δnunwunt^−δn is reducible, i.e., w =u⁻¹_n y_−δ^p

nun for some p∈ Z. Let u⁰ =u_nuu⁻¹_n = u_nu₀t^δ1· · ·t^δn and w⁰ =u_nwu⁻¹_n =y_−δ^p

n. Then u⁰wu⁰⁻¹=w^r, i.e., u⁰ ∈NG^∗

¡hw⁰i¢ .

Ifn= 1, thenu0t^δ1y_−δ^p ₁t^−δ1u⁻¹_o =y^pr_−δ1 i.e,u0y_δ^p₁u⁻¹_o =y_−δ^pr₁. This implies

that ³ [

g∈G

hzi^g´

∩ ³ [

g∈G

hxi^g´ 6=h1i.

But this contradicts our assumption thatxandz have different components.

If n ≥ 2, then t^δn−1un−1y_δ^p_nu⁻¹_n−1t^−δn−1 is reducible, i.e., un−1y_δ^p_nu⁻¹_n−1 = y_−δ^q _n−1 for some q∈Z. By Theorem 1 ((5) and (7)), we get thatδn=−δn−1, q=pandun−1=y^s_δn, s∈Z.

Then the worduwould be reducible att^δn−1un−1t^δn=t^−δny^s_δnt^δn which is a contradiction. WhenceNG^∗

¡hwi¢

=NG

¡hwi¢

andCG^∗(w) =CG(w).

Before we consider the case|w| ≥1, we need two lemmas. The first one is a result of the theory of HNN-extensions.

Lemma 4. LetG^∗=­

G, t; t⁻¹At=B®

be a HNN-extension of G. Letv∈G^∗. Then, there are wordsaandbinG^∗ such thatbis cyclicly reduced,v=aba⁻¹ andaba⁻¹is reduced.

Proof. We prove this by induction on |v|. The cases|v| = 0 and |v| = 1 are trivial.

Suppose |v| ≥ 2 and let v = v0t^²1v1· · ·t^²nvn be reduced. If v is cyclicly reduced we are done; so we can assume that the wordα:=t^²nvnv1t^²1 belongs to G andv=v0t^²1˜v¡

v0t^²1¢₋₁ , where

˜ v=

½ v1t^²2v2· · ·t^²n−1vn−1α, ifn≥3

v1α, ifn= 2.

In the second case take a = v0t^²1, b = v1α. In the first case we can apply induction to ˜v=v1t^²2v2· · ·t^²n−1vn−1αand then ˜v= ˜ab˜a⁻¹is reduced for some bcyclicly reduced.

Takea=v0t^²1˜a. Then, by a length argument,v=aba⁻¹is reduced, . ¤X Remark. Let v, a and b be as in previous lemma. It is clear that for every r∈N,v^r=ab^ra⁻¹ and|v^r|=r|b|+ 2|a|. Therefore

|v^r+1| ≥ |v^r| and |v^r+1|=|v^r|if and only if |b|= 0.

Lemma 5. Let w ∈ G^∗ as in Theorem 3. Then there is a vw ∈ G^∗, inde- composable, such thatw=v_w^r for somer∈N. If w is cyclicly reduced, so is vw.

Proof. First we prove the statement for w cyclicly reduced by induction on

|w|. If |w|= 0, thenCG^∗(w) =CG(w) =hvwi,where vw is an atomic element

of G. Then w = v_w^r for some r ∈ N, and vw is indecomposable, since it is indecomposable in G.

If |w| ≥ 1 and w is decomposable, let w = v^r (r ∈ N, r ≥ 2). Then v is cyclicly reduced; otherwise there are wordsaandbinG^∗such thatbis cyclicly reduced, |a| > 1, v = aba⁻¹ and aba⁻¹ is reduced. But then w =ab^ra⁻¹ is not cyclicly reduced, a contradiction. Therefore |w|=r|v|and |v|<|w|. The required statement follows by induction.

Now if v ∈ G^∗ is arbitrary, v is conjugate to a cyclicly reduced element w: v = uwu⁻¹ with w cyclicly reduced. By the previous argument we find vw ∈ G^∗ indecomposable such that w = v^r_w for some r ∈ N. It follows that v= (uvwu⁻¹)^r and one can easily verifies thatuvwu⁻¹is also indecomposable.

¤X Now we continue the proof of Theorem 3.

Case II. k :=|w| ≥1. By Lemma 4, it is clear that we can assume, without lost of generality, that w is cyclicly reduced. We can also suppose that w is indecomposable; otherwisew=v_w^r, wherer≥2 andvwis cyclicly reduced and indecomposable. Since|vw|<|w|, it follows by induction thatCG^∗(vw) is cyclic andCG^∗(v^q_w) =CG^∗(vw) for everyq∈N^∗.ThenCG^∗(w) =CG^∗(v_w^r) =CG^∗(vw) is cyclic and for everys∈N^∗we have

CG^∗(w^s) =CG^∗(v_w^rs) =CG^∗(vw) =CG^∗(w).

Let w = w0t^δ1w1· · ·t^δk be cyclicly reduced and indecomposable. Let u = u0t^²1u1· · ·t^²nun be a reduced element ofCG^∗(w^r) (r≥1). We show by induc- tion on |u| =n that u∈ hwi. First we prove that u= 1 or n ≥k. arguing for a contradiction, suppose that u 6= 1 and n < k. It follows by induction that CG^∗(u) is cyclic. Let CG^∗(u) = hvui. Then u=v^q_u for some q∈Z^∗ and

|vu| ≤ |u|. It follows that

CG^∗(vu) =CG^∗(v_u^q) =CG^∗(u) =hvui.

By hypothesis,w^r∈CG^∗(u) =hvui.Thenw^r=v_u^±s(s≥1) and w∈CG^∗(w^r) =CG^∗(v_u^±s) =CG^∗(u) =hvui.

Then w =v_u^±p (p≥1). Since w is indecomposable, p= 1 and |w| = |vu| ≤

|u|<|w|, a contradiction.

So we can suppose thatu6= 1 andn≥k. Since u∈CG^∗(w^r) we have that uw^ru⁻¹=w^r. More explicitly:

u0t^²1u1· · ·t^²n−kun−kt^²n−k+1· · ·t^²n−1un−1t^²nunw0t^δ1· · ·

· · ·t^δk· · ·w0t^δ1w1· · ·t^δku⁻¹_n t^−²n· · ·t^−²1u⁻¹₀ =w^r.

We prove that the word from the left either att^²nunw0t^δ1 or att^δku⁻¹_n t^−²n is reducible, but not at both positions. Since wis cyclicly reduced,|w^r|=r|w|.

Then it is clear that the word on the left of the equality above is reducible and the only two positions where this is possible are the ones mentioned above.

Suppose it is reducible at both positions. Then t^²nunw0t^δ1 is reducible, i.e.

²n =−δ1 and w0=u⁻¹_n y^p_δ1 for some p∈Z^∗ andt^δku⁻¹_n t^−²n is also reducible, i.e.,δk=²n=−δ1andu⁻¹_n =y_δ^q₁ for someq∈Z^∗.

Ifk= 1, thenδk =−δ1 is already a contradiction. Ifk >1, thenw would not be cyclicly reduced because

t^δkw0t^δ1 =t^−δ1u⁻¹_n y_δ^p₁t^δ1 =t^−δ1y^p+q_δ1 t^δ1 =y^p+q_−δ1. This is also a contradiction.

To finish the proof we show that in both cases we haveu∈ hwi.

Subcase IIa: The word on the left of the equality above is reducible att^δku⁻¹_n t^−²n. Leti≤k−1 maximal so that

t^δk−iwk−i·...·t^δk−1wk−1t^δku⁻¹_n t^−²n·...·u⁻¹_n−it^−²n−i =y^p_²n−i for somep∈Z^∗. Let

˜

w=t^δk−iwk−i· · ·t^δk−1wk−1t^δk, w¯=w0t^δ1t^δk−i−1wk−i−1

and

¯

u=u0t^²1u1· · ·t^²n−i−1un−i−1, u˜=t^²n−iun−i· · ·t^²nun.

Then ˜u = y_²^−p_n−iw. We show now that˜ i = k−1, i.e., ¯w = w0. Suppose, for a contradiction, that |w| ≥¯ 1. Then ¯wy^p_²n−1u¯⁻¹ is not reducible. But w^r=uw^ru⁻¹=uw^r−1wy¯ _²^p_n−iu¯⁻¹ anduw^r−1w¯ is also reduced. It follows that

r|w|=|u|+ (r−1)|w|+|w|¯ +|¯u|,

i.e., |w|=|u|+|w|¯ +|¯u|>|u|, a contradiction. We then have that w=w0w˜ andu= ¯uy_²^−p_n−iw₀⁻¹w. Let ˆu= ¯uy^−p_²n−iw⁻¹₀ . Thenu= ˆuw and

w^r=uw^ru⁻¹= ˆuww^rw⁻¹uˆ⁻¹= ˆuw^ruˆ⁻¹.

Since|ˆu|<|u|, it follows by induction that ˆu∈ hwi. Thereforeu∈ hwi.

Subcase IIb: The word on the left of the equality above is reducible att^²nu_nw₀t^δ1. Letj≤k−1 maximal, so that

t^²n−jun−j· · · ·t^²n−1un−1t^²nunw0t^δ1w1t^δ2· · · ·wjt^δj+1 =y^q_−δj+1 for someq∈Z^∗. Let

¯

w=w0t^δ1· · · ·wjt^δj+1, w˜=wj+1t^δj+2· · · ·t^δk

and

¯

u=u0t^²1u1· · ·t^²n−j−1un−j−1, u˜=t^²n−jun−j· · · ·t^²nun. Then ˜u =y_−δ^q

j+1w¯⁻¹. We show that j =k−1, i.e., ˜w = 1. Suppose, for a contradiction, that|w| ≥˜ 1, i.e. ¯uy_−δ^q _j+1w˜ is reduced. Then

w^r=uw^ru⁻¹= ¯u˜uw¯ww˜ ^r−1u⁻¹= ¯uy_−δ^q _j+1ww˜ ^r−1u⁻¹ and ˜ww^r−1u⁻¹ is reduced. It follows that

r|w|=|¯u|+|w|˜ + (r−1)|w|+|u|,

i.e.,|w|=|¯u|+|w|+|u|˜ >|u|, a contradiction. Then ¯w=wand ˜u=y_−δ^q _j+1w⁻¹. Let ˆu= ¯uy_−δ^q _j+1. Then

w^r=uw^ru⁻¹= ˆuw⁻¹w^rwˆu⁻¹= ˆuw^ruˆ⁻¹.

Since|ˆu|<|u|, it follows by induction that ˆu∈ hwi. Thenu= ˆuw⁻¹∈ hwi.

(ii) Arguing for contradiction, suppose that C_y^G∗ = C_x^G∗. We may assume, without lost of generality, that y is atomic. Then there is a reduced word u=u0t^²1u1·...·t^²kuk (k≥1) inG^∗, so thatu⁻¹xu=y.

Ifk= 1, theny =u⁻¹₁ t^−²1u⁻¹₀ xu0t^²1u1 is reducible. So u⁻¹₀ xu0 is inhxior inhzi. ButC_x^G∗6=C_z^G∗. Thenu⁻¹₀ xu0∈ hxi. It follows that

u⁻¹₀ xu0=x, u0=x^r0 (r0∈Z) and²1= 1.

Then u⁻¹₁ zu1 = y, a contradiction. Then we can assume that k ≥ 2. So t^−²2u⁻¹₁ t^−²1u⁻¹₀ xu0t^²1u1t^²2 is reducible. It follows that

u0=x^r0 (r0∈Z), ²1= 1, u1=z^r1 (r1∈Z), and²2=−1.

Thent^²1u1t^²2 would be reducible, a contradiction. ¤X

3. Pseudo-bad groups

Now we use Theorem 3 in order to construct a pseudo-bad group.

Definition. A Chain of (∗)-groups ¡ Gi

¢

i∈I is called a(∗)-chain, if for every i, j ∈I, ifi≤j, then Gi ⊆Gj, and for every i∈I, everyu∈Gi and every j≥i,CGj(u) =CGi(u).

Theorem 6. Let¡ Gi

¢

i∈I be a(∗)-chain. ThenG= S

i∈I

Gi is a(∗)-chain and for everyi∈Iand everyu∈Gi,CG(u) =CGi(u).

Proof. Letu∈Gi. It is enough to prove thatCG(u)⊆CGi(u). Letv∈CG(u), thenv∈Gj for somej ≥i. Whencev∈CGj(u) =CGi(u). ¤X

Corollary 7. Every(∗)-group G can be embedded in a(∗)-groupG⁰, so that Gr{1}is totally contained in one component ofG⁰and for everyu∈Gr{1}, C_G⁰(u) =C_G(u).

Proof. Let¡ xα

¢

α<β be an ordering of a representatives system for the compo- nents of G consisting only of atomic elements. Let

G⁰=­ G,¡

tα

¢

α<β−{0}; t⁻¹_α x0tα=xα

® G⁰ = S

α<β

Gα, where G0 := G, Gα+1 :=­

Gα, tα+1; t⁻¹_α+1x0tα+1 =xα+1

® and Gλ = S

α<λ

Gα for someλ a limit ordinal. From Theorem 3 and Theorem 6 it follows that¡

Gα

¢

α<β is a (∗)-chain and so the corollary follows. ¤X The next corollary follows likewise.

Corollary 8. Every(∗)-group G can be embedded in a(∗)-groupG˜ with only one component, i.e.,G˜= S

g∈G˜

CG˜(x)^g for some fixed x.

Proof. Applying successively Corollary 7, we get a (∗)-chain G=G⁽⁰⁾≤G⁰ ≤G⁽²⁾ ≤ · · ·,

where for everyS n, G⁽ⁿ⁾r{1} is contained in one component ofG⁽ⁿ⁺¹⁾. ˜G=

n<ωG⁽ⁿ⁾is a (∗)-group with only one component. ¤X

LetG=F(x, z) be the free group over{x, z}. Gis a (∗)-group. Let ˜Gbe as in Corollary 8 andB =hxi.

Theorem 10. The groupG˜ of Corollary 8 is a pseudo-bad group. Moreover, the maximal solvable subgroups ofG˜ areB and its conjugates.

Proof. By construction ˜Gis a (∗)-group with only one component. By Theorem 1 ((4), (9) and (2)),NG(B) =B and the maximal solvable subgroups of ˜Gare B and its conjugates. Then, all we have to prove is that ˜G is simple. Let N 6={1} be a normal subgroup of ˜G; N contains an a ∈ Br{1}. Without lost of generality, we may assume thata=x^r for somer∈N^∗,rminimal such thatx^r∈N. It is clear that

N = [

g∈G˜

hx^ri^g.

Thenz^r∈N andx^rz^r∈N, i.e.,x^rz^r=¡ u^r¢_n

for some atomic elementuand some n∈Z. Then u∈F(x, z). Since x^rz^r is indecomposable, r· |n|= 1 and sor= 1. We have thenN = ˜G. ¤X

In fact, we have a family of pseudo-bad groups, one for each (∗)-group G we start with. ˜Glooks like a minimal simple bad group, but it is not of finite Morley rank since ˜G contains the definable subgroupB ∼= Z. We even have the following result.

Theorem 11. Let G be a non trivial(∗)-group (e.g.,G=F(x, z)), and letG˜ be like in Corollary 8. ThenG˜ is not superstable.

Proof. We prove the following claim:

Claim. For everyb1,· · ·, bn ∈G, there exists a˜ g∈G, such that˜ b1g,· · ·, bng are not squares inG.˜

G˜ is by construction a union of (∗)-groups Gα’s. Let α0 be such that b1,· · ·, bn ∈ Gα0. Then Gα0+1 = ­

Gα0, tα0+1; t⁻¹_α0+1x0tα0+1 = xα0+1

®, and it is clear that b1tα0+1,· · ·, bntα0+1 are not squares in Gα0+1, neither in ˜G;

otherwisebitα0+1 =u² for someu∈G˜rGα0+1. Then u∈CG˜(u²)⊆Gα0+1, a contradiction.

By the claim and a lemma from [Po 2], it follows that if ˜Gis in fact stable;

then ifais a generic element over ˜G,ais not a square. Soa²is not generic over G. But˜ ais algebraic over a²because ais the only square root ofa². Then ˜G is not superstable since for every super-stable group the following holds: if a generic element is algebraic over an elementb, then bitself is generic. ¤X

References

[B-P] A. Borovik & B. Poizat,Mauvais groupes, Sibirskii Mathemati ˘ceskii ˘Zurnal.

[Ch] G. Cherlin,Bad groups of small Morley rank, Ann. of Math. Logic17(1979), 1–28.

[Co1] L. J. Corredor,Gruppen vonendlichen Morley-Rang, Ph. D. Thesis, Bonn, 1989.

[Co2] L. J. Corredor,Bad groups of finite Morley rank, J. of Symbolic Logic54(1989), 768–773.

[I] S. V. Ivanov,Abstracts of 11th All-Union Symposium on Theory of Groups, Kun- gurka, 1989, p. 51.

[L-Sch]Lyndon, Schupp,Combinatorial Group Theory, Springer-Verlag, 1977.

[O] A. Yu. Ol˘skanskii,Groups of bounded period with subgroups of prime order, Al- gebra i Logika21(1982), 553–618.

[Po 1] B. Poizat,Groupes Stables, Nur al-Mantik wal- Ma’rifah, Villeurbane, 1985.

[Po 2] B. Poizat,Groupes Stables avec types g´en´eriques reguliers, J. of Symbolic Logic48 (1983), 339–355.

(Recibido en julio de 1998)

Departamento de Matematicas Universidad de los Andes Apartado Aereo 4976 Bogota, COLOMBIA

e-mail: [email protected]