Linear Groups of Finite Morley Rank

arXiv:0801.3958v2 [math.LO] 15 Nov 2008 DEFINABLY LINEAR GROUPS OF FINITE MORLEY RANK ALEXANDRE BOROVIK AND JEFFREY BURDGES Introduction. Zilber’s original trichotomy conjecture proposed an explicit classification of all one-dimensional objects arising in model theory. At one point, classifying the simple groups of finite Morley rank was viewed as a subproblem whose affirmative answer would justify this conjecture. Zilber’s conjecture was eventually refuted by Hrushovski [9], and the classification of simple groups of finite Morley rank remains open today. However, these conjectures hold in two significant cases. First, Hrushovski and Zilber prove the full trichotomy conjecture holds under very strong geometric assumptions [10], and this suffices for various diophantine applications. Second, the Even & Mixed Type Theorem [1] shows that simple groups of finite Morley rank containing an infinite elementary abelian 2-subgroup are Chevellay groups over an algebraically closed field of characteristic two. In this paper, we clarify some middle ground between these two results by eliminating involutions from simple groups which are definably embedded in a linear group over an algebraically closed field in a structure of finite Morley rank, and which are not Zariski closed themselves. One may simplify terminology by saying that G is a definably linear group over a field k of finite Morley rank, implicitly using some expansion of the field language, or just a definably linear group of finite Morley rank. Theorem. Let G be a definable infinite simple subgroup of GLn (k) over a field k of finite Morley rank and characteristic zero, which is not Zariski closed. Then G has no involutions, and its Borel subgroups are self-normalizing. Poizat has shown that simple groups with such a definable linear embedding are Chevellay groups when the field has characteristic p > 0 [13]. Much geometric information is available about such groups in characteristic zero but not as much as [10] assumes. For our purposes, the most important geometric fact is that every element of a counterexample is semisimple in the ambient linear group. We note that such groups include those groups interpretable in bad fields, which have been recently shown to exist [3]. A major feature of our proof is the elimination of 2-tori of outer automorphisms of simple groups via the Delehan-Nesin argument [5, Prop. 13.4] (see Lemma 3.3). Such an application is a hopeful sign for the current “L∗ ” project of classifying simple groups of finite Morley rank which have an involution. On the other hand, our proof is driven primarily by one unreasonably strong fact about these linear Supported by NSF postdoctoral fellowship DMS-0503036, a Bourse Chateaubriand postdoctoral fellowship, and DFG grant Te 242/3-1. 1 2 ALEXANDRE BOROVIK AND JEFFREY BURDGES groups: all strongly real elements, i.e. all inverted elements, have a nontrivial power inside a unique conjugacy class of Borel subgroups which are pairwise disjoint. In this vein, we leave the proof of the following consequence as an exercise to the reader (see also [2, 4.19]). Corollary. A definably linear group of finite Morley rank is an L-groups. In the first section, we reduce the problem first to characteristic zero using a result of Poizat, and then to the hypotheses used later by our inductive argument. In the second section, we recall various facts about groups without unipotent torsion. In the third section, we prove our main result using the earlier reductions. The authors thank Oliver Frécon for helpful comments and corrections. §1. Definably linear groups. In this section, we strengthen our hypotheses by recalling some past work on definably linear groups of finite Morley rank. Poizat has already shown that simple linear groups over fields of characteristic p > 0 are isomorphic to algebraic groups. Fact 1.1 (Poizat [13, Theorem 1]). Let k be a field of finite Morley rank and characteristic p > 0. Then any simple definable subgroup of GLn (k) is isomorphic to an algebraic group over some field of characteristic p, but not necessarily k. As such, our theorem considers only fields of characteristic zero. Hypothesis 1.2. Let G be a definable connected subgroup of GLn (k) over a field k of finite Morley rank and characteristic zero. We now show that all definable subgroups satisfy conjugacy of Borel subgroups, i.e. maximal connected definable solvable subgroups, with the following two facts. Fact 1.3 (Poizat [13, Theorem 3]). If such a group G is simple, but is not Zariski closed, then all elements of G are semisimple. Lemma 1.4 (Mustafin [12, Proposition 2.10]). If our group G consists entirely of semisimple elements, then all Borel subgroups of G are abelian and conjugate, and this conjugacy class is generic. Proof. Let B be a Borel subgroup of G. The Zarisk closure B̂ of B is a connected solvable group. So its derived subgroup B̂ ′ consists entirely of unipotent elements. As B consists entirely of semisimple elements, we have B ′ ≤ B ∩ B̂ ′ = 1, and B is abelian. It follows that B̂ is an algebraic torus. Let T be a maximal algebraic torus of GLn (k) which contains B. There are only finitely many distinct intersections T ∩ T g for g ∈ GLn (k), by [7, Rigidity II]. As any distinct conjugates of B lie in distinct conjugates of T , there are only finitely many distinct intersections B ∩B g for g ∈ G too, and so the union of such intersections is not generic in B. By the genericity argument [4, Lemma 4.1], S B G is generic in G. Now G has only one conjugacy class of Borel subgroups, by [11]. ⊣ DEFINABLY LINEAR GROUPS OF FINITE MORLEY RANK 3 Proposition 1.5. If G is simple, but not Zariski closed, then Borel subgroups of G are abelian and hereditarily conjugate. Proof. By Fact 1.3, all elements of G are semisimple. As this property passes to subgroups, the result follows from Lemma 1.4. ⊣ It follows that our G satisfies Hypothesis 3.1 below. So then Theorem 3.8 will show that G has no involutions. We recall that a Carter subgroup of G is an almost self-normalizing definable nilpotent subgroup of G In our case, these are merely our Borel subgroups. As the Borel subgroups are conjugate, the Borel subgroups of G will then be selfnormalizing too, by Lemma 2.3. So these two results will complete the proof. §2. No unipotent torsion. We now recall several convenient results from [6] concerning groups without unipotent torsion, i.e. which contains no definable connected nilpotent subgroup of bounded exponent. Lemma 2.1. Let G be a connected group of finite Morley rank without unipotent torsion. Then every element has a nontrivial power which lies inside a Borel subgroup. This is a corollary of the following fact. Fact 2.2 ([6, Theorem 3]). Let G be a connected group of finite Morley rank of π ⊥ -type, i.e. without unipotent p-torsion for p ∈ π. Then all π-elements of G are toral, i.e. lie inside π-tori. We recall that, in this setting, divisible torsion groups are referred to as tori and their elements may be referred to as toral. As usual, H ◦ denotes the connected component of H, so a Sylow◦ 2-subgroup is just a maximal 2-torus. Proof of Lemma 2.1. Consider an element a ∈ G. If d(a) is infinite, we choose n = |d(a)/d(a)◦ |. If d(a) is finite, a lies inside some torus by Fact 2.2. ⊣ We observe that, if one assumes that Borel subgroups are conjugate, one may instead prove that all elements lie inside Borel subgroups. For this, we require that the group has degenerate type, meaning a finite Sylow 2-subgroup or equivalently no involutions [4, Theorem 1]. Lemma 2.3. Let G be a connected degenerate type group of finite Morley rank without unipotent torsion. Then G has some self-normalizing Carter subgroup. This is a corollary of the following. Definition 2.4. Let G be a group of finite Morley rank, and T a maximal divisible abelian torsion subgroup of G. The Weyl group of G is the finite group N (T )/C ◦ (T ), which can be viewed as a group of automorphisms of T . Fact 2.5 ([6, Thm. 5]). Let G be a connected group of finite Morley rank. Suppose the Weyl group is nontrivial and has odd order, with r the smallest prime divisor of its order. Then G contains a unipotent r-subgroup. Fact 2.6 ([6, Lemma 1.2]). Let G be a connected group of finite Morley rank, T a maximal p-torus in G, and suppose that T is central in G. If G has p⊥ -type, then any p-element a ∈ G belongs to T . 4 ALEXANDRE BOROVIK AND JEFFREY BURDGES Proof of Lemma 2.3. We may assume that G contains torsion. Let T be a maximal divisible abelian torsion subgroup of G. By [8], G has a Carter subgroup ◦ ◦ Q containing T . By Fact 2.5, NG (T ) = CG (T ). So NG (Q) ≤ CG (T ). By Fact ◦ 2.6, CG (T )/d(T ) is torsion free. So NG (Q) ≤ Q, as desired. ⊣ §3. Bad-like groups. Here we analyze groups satisfying the following hypotheses. Hypothesis 3.1. Let G be a connected group of finite Morley rank without unipotent torsion whose Borel subgroups are nilpotent and hereditarily conjugate, i.e. any subgroup H of G satisfies H-conjugacy of Borel subgroups of H. These hypotheses clearly pass to subgroups. We observe that they also pass to sections by nilpotent kernels. Lemma 3.2. Let K be a nilpotent normal subgroup of G. Then G/K satisfies Hypothesis 3.1 too, and images and inverse images preserve Borel subgroups. Proof. Quotients and extensions of solvable groups are solvable, so images and inverse images preserve maximal solvable subgroups. As images also preserve connectivity, they preserve Borel subgroups, and Borel subgroups of G/K are nilpotent. If K is connected, inverse images are connected too, and hence preserve Borel subgroups too. We may now assume that K is finite, and even cyclic of order pn . As H is connected, K is central in G. So, by Fact 2.2, K is contained in every maximal p-torus of G. By conjugacy, K is contained in every Borel subgroup of G too. So all inverse images are connected, and hence preserve Borel subgroups. Our first part now follows easily by applying the second inside all definable connected subgroups of G ⊣ A variation on the Delehan-Nesin argument [5, Prop. 13.4] shows that 2-tori have no interesting actions inside such a group. Proposition 3.3. Let H be a definable normal subgroup of G which has no involutions. Then every 2-torus in G centralizes H. For this, we note several facts about such a degenerate type group H normalized by an involution. Fact 3.4. No element a ∈ H is H-conjugate to it’s inverse a−1 . Proof. Suppose that ah = a−1 for some a, h ∈ H. As a is not an involution, we have h ∈ NH (d(a)) \ CH (a). But clearly h has order two modulo CH (a). So H contains an involution by [5, Ex. 11 p. 93; Ex. 13c p. 72], a contradiction. ⊣ Lemma 3.5. No element of CH (i) is inverted by any involution in hiiH. Proof. Any involution of hiiH is conjugate to i under H. So otherwise there is an x ∈ X − such that xh ∈ C(i). We may assume that h ∈ X − too because [5, Ex. 14 p. 73] says H = XCH (i) where X := {g ∈ H | g i = g −1 }. So −1 2 xh = (xh )i = hx−1 h−1 = (x−1 )h , then xh = x−1 , contradicting Fact 3.4. ⊣ Proof of Proposition 3.3. Consider a 2-torus T of G. We may take G := Hd(T ). We may assume G is centerless, by Lemma 3.2. So there is a nontrivial strongly real element ij inside H. It follows, by Lemma 2.1, that some nontrivial DEFINABLY LINEAR GROUPS OF FINITE MORLEY RANK 5 power x of ij lies inside some Borel subgroup B of H. As all Borel subgroups of G are conjugate, some conjugate T h of T lies inside the Borel subgroup of G containing B. But now T h centralizes B by [5, Theorem 6.16], contradicting Lemma 3.5. ⊣ We say a subgroup K of a group H is subnormal if there is a finite chain K ✁ K1 ✁ · · · ✁ Kk ✁ H. A quasisimple subnormal subgroup of a group H is referred to as a component of H. If H is connected, then its components are definable, connected, and normal in H, by [5, Lemma 7.10]. They also centralize one another by [5, Lemma 7.12]. We let E(H) denote the central product of the components in H, which happen to be connected, and set F ∗ (H) := F ◦ (H)E(H). According to the following, Aut(F ∗ (Ḡ)) controls the structure of Ḡ. Fact 3.6 ([5, Thm. 7.13 & Cor. 7.14]). Any group H of finite Morley rank satisfies CH (F ∗ (H)) = Z(F ∗ (H)) ≤ F ∗ (H) and H/Z(F ∗ (H)) ≤ Aut(F ∗ (H)). Lemma 3.7. If all simple sections of G with nilpotent kernels have degenerate type, then G has a unique Sylow 2-subgroup, which is connected and central. Proof. A connected degenerate type group of finite Morley rank has no involutions by [4, Theorem 1]. So, as E(G)/Z(E(G)) is 2⊥ by hypothesis, we find that E(G) is 2⊥ too. By Lemma 3.3, the Sylow◦ 2-subgroup S ◦ of G centralizes E(G). But G ֒→ Aut(E(G)) by Fact 3.6. So any 2-torsion in G is lies inside the solvable radical σ(G), which lies inside F (G) because Borel subgroups are nilpotent. It follows that the Sylow 2-subgroup is connected by Fact 2.2 and [5, Theorem 9.29], and central by [5, Theorem 6.16]. ⊣ We now prove our main result. Theorem 3.8. Suppose that G is simple. Then G has no involutions. Proof. We consider a counterexample G of minimal Morley rank. So, by Lemma 3.2, all proper simple section with nilpotent kernels have degenerate type. By Lemma 3.7, any proper connected subgroup H centralizes its Sylow 2-subgroup, which is a 2-torus. As G is simple, it follows that (⋆1 ) the centralizers of distinct Sylow◦ 2-subgroups are disjoint. Also, any Sylow◦ 2-subgroup is central in a Borel subgroup. By conjugacy, all Borel subgroups have a central Sylow◦ 2-subgroup. So two Borel subgroups meet iff they both contain the same Sylow◦ (⋆2 ) 2-subgroup of G. Claim 3.9. Any two distinct involutions i and j normalize a common Sylow◦ ◦ 2-subgroup T of G, and CG (T ). Verification. By Lemma 2.1, there is a nontrivial power x of ij inside a Borel subgroup Q of G. As i inverts x, Qi also contains x. By ⋆2 , Q and Qi contain the same Sylow◦ 2-subgroup T of G. So i normalizes T . Similarly, j normalizes T too. ✸ 6 ALEXANDRE BOROVIK AND JEFFREY BURDGES Every involution is contained in a Sylow◦ 2-subgroup, by Fact 2.2, which is unique by ⋆1 . Let Ti denote the Sylow◦ 2-subgroup containing i ∈ I(G). Now ◦ ◦ ◦ CG (i) = CG (Ti ) since Ti is the unique Sylow◦ 2-subgroup of CG (i), by ⋆1 . Claim 3.10. Let T be a Sylow◦ 2-subgroup of G, and let j be an involution ◦ normalizing T . Then either j ∈ T or j inverts CG (T ). In particular, CG (T ) is an abelian Borel subgroup of G. Verification. Suppose that j ∈ / T . Then CG (T, j) = CG (Tj ) ∩ CG (T ) = 1. ◦ ◦ So, by [5, Ex. 13 & 15, p. 79], CG (T ) is abelian and inverted by j. Now CG (T ) is a Borel subgroup of G, as it containes a Borel subgroup of G. ✸ As a consequence, if involutions i and j do not commute, then neither i nor j lie inside a Sylow◦ 2-subgroup which they both normalize, and both invert its Borel subgroup. Claim 3.11. G has Prüfer rank one. So a Sylow 2-subgroup of G is isomorphic to that of PSL2 . Verification. Suppose towards a contradiction that pr(G) > 1. Choose a pair of non-commuting involutions i and j, and fix a Sylow◦ 2-subgroup T which ◦ is normalized by them both. As i inverts CG (T ) by Claim 3.10, Ω1 (T ) normalizes ◦ ◦ ◦ CG (Ti ). By generation [6, Theorem 1], CG (Ti ) = hCG (Ti , u) | u ∈ I(T )i. So ◦ CG (Ti , u) 6= 1 for some u ∈ I(T ), which yields the contradiction T = Ti . Thus pr(G) = 1. Let S be a Sylow 2-subgroup of G. By Fact 2.2, CS (S ◦ ) = S ◦ , and [S, S ◦ ] ≤ 2. There are involutions in S \ S ◦ by Claims 3.9 and 3.10. So S has the desired structure. ✸ Consider the set I of all involutions in G. We define the set L of lines on I to be the collection of cosets jQ where Q is a Borel subgroup of G and j inverts Q. We now prove that I and L form a projective plane. PP2. Any two points lie on a unique line. Any two involutions i, j ∈ I normalize some Sylow◦ 2-subgroup T , by Claim ◦ 3.9. By Claim 3.10, Q := CG (T ) is a Borel subgroup inverted by i and j. So i and j lie on the line jQ. Let x denote the power of ij lies inside some Qg . By ◦ ⋆1 , T g is the unique Sylow◦ 2-subgroup of CG (x). So the line jQ is unique. PP3. Any two lines intersect at a unique point. For any two lines i1 Q1 and i2 Q2 , we consider the involutions j1 ∈ I(Q1 ) and j2 ∈ I(Q2 ), which are unique by Claim 3.11. Again, there is a Borel subgroup R inverted by both, by Claims 3.9 and 3.10. As j1 and j2 both centralize the involution t ∈ I(R), t inverts both Q1 and Q2 , by Claims 3.10 and 3.11. Thus the lines i1 Q1 and i2 Q2 meet at t. Such a point is unique by PP2. PP4. There are four points, no three of which are colinear. Consider a Borel subgroup Q of G, and the unique involution i ∈ I(Q). Let x, y, and z be three involutions inverting Q, and let j be a third point on the line through i and z. The original points i, x, y are obviously not colinear. The DEFINABLY LINEAR GROUPS OF FINITE MORLEY RANK 7 pair i, j is not colinear with either x or y by PP3. Also j, x, y is not colinear by PP2. So i, j, x, y are the four desired involutions. We prove one additional property of this projective plane. PP+. Any three involutions x, y, z ∈ I(G) are colinear iff their product xyz is an involution. The product of three colinear involutions is an involution by Claim 3.11. Consider three involutions x, y, z ∈ I(G) whose product xyz is an involution i. Let Q be the Borel subgroup such that y and z invert Q. As xyz is an involution, (yz)x = (yz)−1 . So x normalizes Q by ⋆1 . But x ∈ / Q as Q has only one involution. So x inverts Q, and x lies on the line with y and z. These four conditions contradict the following theorem. Bachmann’s Theorem ([5, Theorems 8.15 & 8.18]). Let G be a group whose set of involutions I posses the structure of a projective plane, and three involutions are colinear iff their product is an involution. Then hIi ∼ = SO3 (k, f ) for some interpretable field k and some non-isotropic quadratic form f . In particular, G does not have finite Morley rank. This concludes the proof of Theorem 3.8. ⊣ We conclude by conjecturing a a stronger variation the above style of argument exploiting Bachmann’s Theorem. Conjecture 1. There is no simple group of finite Morley rank in which every strongly real elements lies inside some C ◦ (T ) for T is a Sylow◦ 2-subgroup. REFERENCES [1] Tuna Altınel, Alexandre Borovik, and Gregory Cherlin, Simple groups of finite morley rank, Mathematical monographs, Amer. Math. Soc., 2008. [2] Tuna Altınel and Gregory Cherlin, On groups of finite Morley rank of even type, J. Algebra, vol. 264 (2003), no. 1, pp. 155–185. [3] Andreas Baudisch, Martin Hils, Amador Martin-Pizarro, and Frank O. Wagner, Die Böse Farbe, J. Inst. Math. Jussieu, (2008). [4] Alexandre Borovik, Jeffrey Burdges, and Gregory Cherlin, Involutions in groups of finite Morley rank of degenerate type, Selecta Math., vol. 13 (2007), no. 1, pp. 1–22. [5] Alexandre Borovik and Ali Nesin, Groups of finite Morley rank, The Clarendon Press Oxford University Press, New York, 1994, Oxford Science Publications. [6] Jeffrey Burdges and Gregory Cherlin, Semisimple torsion in groups of finite morley rank, Submitted, 2008. [7] Gregory Cherlin, Good tori in groups of finite Morley rank, J. Group Theory, vol. 8 (2005), pp. 613–621. [8] Olivier Frécon and Eric Jaligot, The existence of Carter subgroups in groups of finite Morley rank, J. Group Theory, vol. 8 (2005), pp. 623–633. [9] Ehud Hrushovski, A new strongly minimal set, Ann. Pure Appl. Logic, vol. 62 (1993), no. 2, pp. 147–166, Stability in model theory, III (Trento, 1991). [10] Ehud Hrushovski and Boris Zilber, Zariski geometries, J. Amer. Math. Soc., vol. 9 (1996), no. 1, pp. 1–56. [11] Eric Jaligot, Generix never gives up, J. Symbolic Logic, vol. 71 (2006), no. 2, pp. 599–610. [12] Yerulan Mustafin, Structure des groupes linéaires définissables dans un corps de rang de Morley fini, J. Algebra, vol. 281 (2004), no. 2, pp. 753–773. 8 ALEXANDRE BOROVIK AND JEFFREY BURDGES [13] Bruno Poizat, Quelques modestes remarques à propos d’une conséquence inattendue d’un résultat surprenant de Monsieur Frank Olaf Wagner, J. Symbolic Logic, vol. 66 (2001), no. 4, pp. 1637–1646. SCHOOL OF MATHEMATICS UNIVERSITY OF MANCHESTER OXFORD ROAD M13 9PL MANCHESTER ENGLAND SCHOOL OF MATHEMATICS UNIVERSITY OF MANCHESTER OXFORD ROAD M13 9PL MANCHESTER ENGLAND