Monatsh Math (2009) 157:387–401 DOI 10.1007/s00605-008-0034-6
Sequences of words characterizing finite solvable groups Evija Ribnere
Received: 13 December 2007 / Accepted: 4 March 2008 / Published online: 7 August 2008 © Springer-Verlag 2008
Abstract There are two sequences in two variables which characterize the solvability of finite groups. Namely, the sequence of Bandman, Greuel, Grunewald, −1 Kunyavskii, Pfister and Plotkin which is defined by u 1 = x −2 y −1 x and u n = [xu−1 n−1 x , −1 −1 yu n−1 y ] and the sequence of Bray, Wilson, and Wilson defined by s1 = x and −y sn = [sn−1 , sn−1 ]. We define new sequences and proof that six of them characterize the solvability of finite groups. Keywords Finite solvable group · Identity · Simple group · Gröbner basis · Engel group Mathematics Subject Classification (2000)
20D10 · 20E10 · 20D05
1 Introduction A group G is nilpotent by definition if the identity [[[x0 , x1 ], x2 ], . . . , xn ] = 1 holds for some n ∈ N and all x0 , x1 , . . . , xn ∈ G. We use the following notation for commutators and conjugates: [x, y] = x −1 y −1 x y, x y = x −1 yx, x −y = x −1 y −1 x. In 1936, Zorn [16] showed that the nilpotency of a finite group can be expressed by an identity involving only two variables. Namely, a finite group G is nilpotent if and only if it satisfies the Engel-identity en (x, y) = 1 for some n ∈ N and all x, y ∈ G. Here e1 (x, y) = [x, y] and en (x, y) is inductively defined by
Communicated by D. Segal. E. Ribnere (B) Universität Düsseldorf Heinrich-Heine-Universität, Universitätsstrasse 1, 40225 Düsseldorf, Germany e-mail:
[email protected]
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en (x, y) = [en−1 (x, y), y]. Huppert [9, III.6.1] showed in his book that there are infinitely many other sequences in two variables which characterize finite nilpotent groups. Similarly, the usual definition of solvability of groups involves many variables. For a long time it was an open question whether solvability can be described by an identity in only two variables. In 2003, Bandman et al. [2] found such a sequence: a finite group G is solvable if and only if it satisfies the identity u n (x, y) = 1 for some n ∈ N and all x, y ∈ G. Here u n (x, y) is defined by u 1 (x, y) = x −2 y −1 x and u n (x, y) = [xu n−1 (x, y)−1 x −1 , yu n−1 (x, y)−1 y −1 ]. In 2004, Bray et al. [4] found a very nice and short sequence, namely s1 (x, y) = x and sn (x, y) = [sn−1 (x, y)−y , sn−1 (x, y)], and proved an analogous result using group theoretical methods. In this paper, we define new sequences and show that six of them characterize finite solvable groups as well. Definition 1.1 Let f, g, h be words in {x, y, x −1 , y −1 }. We define a sequence (v1 , v2 , . . . ) by setting g h . v1 := f and vk := vk−1 , vk−1 We can plug any pair (x, y) ∈ G 2 of group elements into (v1 , v2 , . . . ) to get a sequence (v1 (x, y), v2 (x, y), . . . ) in G. Note that every vn+1 (x, y) lies in the nth term of the derived series of G. Hence, if G is solvable with derived length n 0 , then vn (x, y) = 1 for all n > n 0 . Here we prove the following result. Theorem 1.2 Let G be a finite sequences: −1 −1 y x 1) v1 := yx 2 , vk := vk−1 , −1 −1 y x 2) v1 := yx 2 , vk := vk−1 , −1 −1 yx y 3) v1 := x y, vk := vk−1 , −1 −1 yx y 4) v1 := x y, vk := vk−1 , −1 −1 yx y 5) v1 := x y, vk := vk−1 , −1 −1 yx y 6) v1 := x y, vk := vk−1 ,
group and (v1 , v2 , . . . ) one of the following six x −1 , vk−1 yx vk−1 , x , vk−1 y −1 vk−1 , x yx vk−1 , y −1 x −1 y −1
vk−1
.
If there is an n ∈ N with vn (x, y) = 1 for all x, y ∈ G then G is solvable. By the above remark the converse holds as well. Moreover, we will show that there are many other sequences for which an analogous theorem can be proved.
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2 Proof of the Theorem 1.2 The main strategy of the proof is the same as in [2,3], and [4], as we prove the converse: If G is non-solvable, then there are x, y ∈ G such that vn (x, y) = 1 for all n ∈ N. Let G be a minimal counter example, i.e., a finite non-solvable group of the smallest order with vn (x, y) = 1 for some n ∈ N and all x, y ∈ G. Then G must be simple and must have only solvable subgroups, because any identity remains true in subgroups and quotients. By Thompson [15] G is one of the groups: PSL(3, F3 ), PSL(2, Fq ) (where q ≥ 4, q = p n , p a prime) or Sz(2n ) (where n ∈ N , n ≥ 3 and odd). It is enough to show: For any of the above groups there are elements x and y with vn (x, y) = 1 for all n ∈ N. The idea of [2] is to prove that there are x, y ∈ G such that v1 (x, y) = 1 and v1 (x, y) = v2 (x, y). This implies immediately 1 = v1 (x, y) = vn (x, y) for all n ∈ N. Using this it is sufficient to prove the next lemma: Lemma 2.1 Let vn be one of the six sequences of Theorem 1.2 and G one of the following groups: PSL(3, F3 ), PSL(2, Fq ) (where q ≥ 4, q = p n , p a prime), or Sz(2n ) (where n ∈ N , n ≥ 3 and odd). Then there are x, y ∈ G such that v1 (x, y) = v2 (x, y) and v1 (x, y) = 1 . The proof for the sequences 3), 4), 5) and 6) follows directly from [2]. It is easy to check v1 (x, y) = 1 for x, y which were used in [2]. For these x and y the second condition v1 (x, y) = v2 (x, y) holds as well, because v1 (x, y) = v2 (x, y) is exactly the same identity as u 1 (x, y) = u 2 (x, y). The equality v1 (x, y) = v2 (x, y) is equivalent for the sequences 1) and 2). Thus, we only need to prove the lemma for the first sequence: −1 −1 y x x −1 v1 := yx 2 , vk := vk−1 , vk−1 . For the proof we follow the main steps of [2], however some steps are simplified and more explicit, in particular in the Sz(2n ) case. 2.1 Case PSL(3, F3 ) We take ⎛
⎞ 0 0 1 ⎜ ⎟ x := ⎝ 2 2 0 ⎠, 0 2 0
⎛
⎞ 0 0 2 ⎜ ⎟ y := ⎝ 2 2 0 ⎠, 0 1 1
and compute ⎛
⎞ ⎛ ⎞ 2 1 0 2 1 0 ⎜ ⎟ ⎜ ⎟ v1 (x, y) = ⎝ 0 0 1 ⎠, v2 (x, y) = ⎝ 0 0 1 ⎠, 1 0 2 1 0 2 which proves the first case of Lemma 2.1.
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2.2 Case PSL(2, Fq ) Let p be a prime and q = p k for some k ∈ N. We use the ansatz as in [2]:
t −1 1 a x(t) := and y(a, b) := ∈ M(2, Z[t, a, b]). 1 0 b 1 + ab At first, we note that v1 (x, y) = 1 is equivalent to yx 2 = 1 and v1 (x, y) = v2 (x, y) is equivalent to y −2 x −2 yx = x 2 y −1 x −1 y. We set
m 11 m 12 := y(a, b)−2 x(t)−2 y(a, b)x(t) − x(t)2 y(a, b)−1 x(t)−1 y(a, b), m 21 m 22
n 11 n 12 n 21 n 22
:= y(a, b)x(t) − 2
=
1 0 0 1
t 2 + ta − 2 2 t b + tab + t − b
−t − a , −tb − ab − 2
and define I := m 11 , m 12 , m 21 , m 22 ,
I0 := n 11 , n 12 , n 21 , n 22
as ideals in Z[t, a, b]. Reduction modulo p yields ideals I , I0 in Fq [t, a, b]. We denote the corresponding algebraic sets in the affine space Fq3 by Vq (I ) and Vq (I0 ). If we find (t, a, b) ∈ Vq (I )\ Vq (I0 ) then we will have x(t) and y(a, b) in PSL(2, Fq ) that satisfy the conditions v1 (x, y) = 1 and v1 (x, y) = v2 (x, y). We consider the ideals I and I0 and obtain that dim V (I ) = 1 over Q3 , so V (I ) is a curve. The complete computations (SINGULAR or MAGMA) can be found in [13]. To make the situation easier we compute the radical ideal J of I over Q[t, a, b] and check I ⊆ J over Z[t, a, b]. Then trivially Vq (J ) ⊆ Vq (I ). Moreover, Vq (I ) \ Vq (I0 ) = ∅ is implied by Vq (J ) ∩ Vq (I0 ) = ∅ and Vq (J ) = ∅. The first is an easy computation, for the second we consider the generators of J : J = (a − b)t + 1, (b2 + 1)t + a 3 b2 + a 3 − a 2 b3 + 3a 2 b − 3ab2 + 5a − b3 − 6b, a 4 b2 + a 4 − 2a 3 b3 + 2a 3 b + a 2 b4 − 6a 2 b2 + 5a 2 + 2ab3 −11ab + b4 + 5b2 − 1. The last generator is a polynomial containing only the variables a and b, and the other two generators are linear in t. Let C := V (a 4 b2 + a 4 − 2a 3 b3 + 2a 3 b + a 2 b4 −6a 2 b2 + 5a 2 + 2ab3 − 11ab + b4 + 5b2 − 1) be the variety of the last generator of J . C is a curve in Q2 . Analogously, we denote by Cq the reduced curve in Fq2 and by C q the projective closure of Cq , which is given by
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the homogenous polynomial F(h, a, b) = −h 6 + 5h 4 a 2 − 11h 4 ab + 5h 4 b2 + h 2 a 4 + 2h 2 a 3 b − 6h 2 a 2 b2 + 2h 2 ab3 + h 2 b4 + a 4 b2 − 2a 3 b3 + a 2 b4 . Lemma 2.2 |Vq (J )| ≥ 1 if |C q | > 9 for any q. Proof There are at most six points (a, b) ∈ Cq such that b2 + 1 = 0 or a − b = 0. Further |C q \Cq | = |C q ∩{(0 : a : b) ∈ P2 }| = |V (F(0, a, b))| = |V (a 2 b2 (a−b)2 )| = |{(0 : 0 : 1), (0 : 1 : 0), (0 : 1 : 1)}| = 3. Then if C q has at least nine points then Cq has at least six points. At least one of them satisfies b2 = 1 or a = b which gives us t. We only have to prove that the curve C q has more than nine points. To estimate the number of points of C q we will use (as in [2]) the Hasse–Weil Theorem: Theorem 2.3 Let C q ⊆ Pn be an absolutely irreducible curve over Fq and Nq the number of its rational points. Then Nq ≥ (q + 1) − 2 pa
√
p,
where pa is the arithmetic genus of C q . To apply the Theorem 2.3 we have to show that C q is irreducible for all q ≥ 4, and compute the genus of C q . This gives us an estimate for the number of rational points of C q if q is large enough. |C q | for small q will be computed by MAGMA directly. 2.2.1 Irreducibility of C q We will prove this using Bezout’s theorem. Assume that C q is reducible. Then there are curves R = V (r (h, a, b)) and S = V (s(h, a, b)) where r (h, a, b), s(h, a, b) ∈ Fq [h, a, b] with no common component such that C q = R ∪ S. Let {P1 , . . . , Pn } be the intersection points of R and S. By Bezout’s theorem [8, 7.8] we have n
i(R, S; P j ) = deg(r ) · deg(s),
j=1
where i(R, S; P j ) denotes the intersection multiplicity of R and S at P j . C q is a variety defined by a polynomial of degree 6, then the possible values of deg(r ) · deg(s) are 1 · 5 = 5, 2 · 4 = 8, or 3 · 3 = 9. The intersection points are always singular points of C q . We will compute the intersection multiplicities of R and S at the singular points of C q and show that their sums can never be equal to 5, 8, or 9. Lemma 2.4 Let p be a prime ∈ / {37, 431} and q = p k . Then the curve C q has 3 singular points P1 = (0 : 1 : 0), P2 = (0 : 1 : 1), P3 = (0 : 0 : 1) over Fq .
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Proof At first we look for singular points in the affine patch U0 := {(1 : a : b) ∈ , ∂ F(1,a,b) in Z[a, b] has Groebner P2 (Fq )}. The ideal D = F(1, a, b), ∂ F(1,a,b) ∂a ∂b 3 4 2 basis a + 26b + 10024b, 2b + 51b + 866, 74b2 + 12580, 15947. The number 15947 = 37 · 431 has an inverse in Fq if q is not a power of 37 or 431, so 1 ∈ D. Therefore, C q has no singular points on U0 . From the proof of Lemma 2.2 we know that C q ∩ {(0 : a : b) ∈ P2 (Fq )} consists of the points {(0 : 0 : 1), (0 : 1 : 0), (0 : 1 : 1)}. These are all singular points by a trivial computation. Lemma 2.5 Let p be a prime ∈ / {37, 431}, q = p k and P1 = (0 : 1 : 0), P2 = (0 : 1 : 1), P3 = (0 : 0 : 1) the singular points of C q = R ∪ S. Then the possibilities for the intersection multiplicities of R and S at P j are the following: 1. i(R, S; P1 ) ∈ {0, 1}, i(R, S; P2 ) ∈ {0, 2}, i(R, S; P3 ) ∈ {0, 1} for p = 2, 2. i(R, S; P1 ) = 0, i(R, S; P2 ) ∈ {0, 4}, and i(R, S; P3 ) = 0 for p = 2. Proof We use Noether’s formula for computing the intersection multiplicities [5, 12.4.2]: i(R, S; P j ) =
m Q (R)m Q (S),
Q
where Q runs over the infinitely near points of P j and m Q are their multiplicities. (1) Let p = 2. We consider the affine patch U1 = {(h : 1 : b) ∈ P2 (Fq )} which contains P1 = (0, 0) and P2 = (0, 1). Let C := C q ∩ U1 = V (−h 6 + 5h 4 − 11h 4 b + 5h 4 b2 + h 2 + 2h 2 b − 6h 2 b2 + 2h 2 b3 + h 2 b4 + b2 − 2b3 + b4 ). We study C in 2 2 P1 . Here √ cone is given by the lowest homogenous component: h + b = √ the tangent (h − −1b)(h + −1b). If p = 2, it has two different factors. Hence the curve C q has two different tangents at P1 , so P1 is an ordinary double point. We get i(R, S; P1 ) = 0 if both branches belong to R (or S) otherwise i(R, S; P1 ) = 1. To study C at P2 we move P2 to the origin by the transformation h → h, b → b +1 and get C = V (−h 6 +5h 4 b2 −h 4 b−h 4 +h 2 b4 +6h 2 b3 +6h 2 b2 +b4 +2b3 +b2 ). Thus C has the double tangent V (b2 ) as tangent cone at P2 . Let C (1) = V (b4 h 4 + b4 h 2 + 6b3 h 3 + 2b3 h + 5b2 h 4 + 6b2 h 2 + b2 − bh 3 − h 4 − h 2 ) be the strict transform under the blow up (h → h, b → hb). The exceptional divisor is E (1) = V (h) and P2(1) = E (1) ∩ C (1) = (0, 0). The curve C (1) has (b2 − h 2 ) = (b − h)(b + h) as tangent cone. (1) Therefore, P2 is an ordinary double point of C (1) , and the curve C q has two branches at the point P2 . If they belong to the same component (R or S), then i(R, S; P2 ) = 0. Assume that they are in different components, so P2 ∈ R ∩ S. Since 2 = m P2 (C q ) = m P2 (R ∩ S) = m P2 (R) + m P2 (S) and m P2 (R), m P2 (S) ≥ 1, we have m P2 (R) = m P2 (S) = 1. Hence, the components R and S are smooth in P2 . Analogously we obtain m P (1) (R (1) ) = m P (1) (S (1) ) = 1 for the infinitely near point P (1) . By Noether’s formula 2
2
we have i(R, S; P2 ) = m P2 (R)m P2 (S) + m P (1) (R (1) )m P (1) (S (1) ) = 1 · 1 + 1 · 1 = 2. 2
2
At last, we consider the affine patch U2 = {(h : a : 1) ∈ P2 (Fq )}, which contains P3 = (0, 0). Then C q ∩ U1 = V (−h 6 + 5h 4 a 2 − 11h 4 a + 5h 4 + h 2 a 4 + 2h 2 a 3 − 6h 2 a 2 + 2h 2 a + h 2 + a 4 − 2a 3 + a 2 ), and this case is analogous to the case P1 .
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(2) Let p = 2. We consider the affine patch U1 = {(h : 1 : b) ∈ P2 (F2 )}, which contains P1 = (0, 0) and P2 = (0, 1). Let C := C 2k ∩ U1 = V (h 6 + h 4 + h 4 b + h 4 b2 + h 2 + h 2 b4 + b2 + b4 ). Its tangent cone at P1 is the double tangent h 2 + b2 = (h + b)2 . The first blow up (h → h, b → hb) gives us the strict transform C (1) = V (h 4 + h 2 + h 3 b + h 4 b2 + 1 + h 4 b4 + b2 + h 2 b4 ), the exceptional divisor E (1) = V (h) and P1(1) = E (1) ∩ C (1) = (0, 1) as the only infinitely near point. We transform (0, 1) to the origin by substitute h → h, b → b + 1 and get C (1) = V (h 2 b4 +h 3 b+h 4 b2 +b2 +h 4 +h 3 +h 4 b4 ). The tangent cone of C (1) in P1(1) is again a double tangent, namely V (b2 ). The second blow up (h → h, b → hb) gives us the strict transform C (2) = V (h 4 b4 + h 2 b + h 4 b2 + b2 + h 2 + h + h 6 b4 ), E (2) = V (h) and P1(2) = E (2) ∩ C (2) = (0, 0) as the only infinitely near point. The curve C (2) is smooth in (0, 0). Thus C 2k is unibranched at P1 , therefore, i(R, S; P1 ) = 0. We consider P2 = (0, 1). After the transformation of (0, 1) to (0, 0) (h → h, b → b + 1) we have C = V (h 2 b4 + h 4 b + h 4 b2 + b4 + b2 + h 6 + h 4 ). Again C has a double tangent V (b2 ) as tangent cone in P2 . The first blow up (h → h, b → hb) gives C (1) = V (h 4 b4+h 3 b+h 4 b2+h 2 b4+b2+h 4+h 2 ), the exceptional divisor E (1) = V (h), (1) and infinitely near point P2 = E (1) ∩ C (1) = (0, 0). The tangent cone of C (1) is again a double tangent given by b2 + h 2 = (b + h)2 . The second blow up gives us C (2) = V (h 4 b6+h 3 b2+h 4 b4+h 2 b4+1+h 4 b2+h 2 ), the exceptional divisor E (2) = V (b) (2) and infinitely near point P2 = E (2) ∩ C (2) = (1, 0). After the transformation (1, 0) → (0, 0) we have C (2) = V (h 2 b4 +h 4 b6 +h 4 b2 +h 2 b2 +h 2 +b6 +h 4 b4 +h 3 b2 +hb2 ) (2) which has a double tangent V (h 2 ) at P2 . The third blow up gives the strict transform C (3) = V (h 2 b4 + h 4 b8 + h 4 b4 + h 2 b2 + h 2 + b4 + h 4 b6 + h 3 b3 + hb), the exceptional divisor E (3) = V (b), and infinitely near point P2(3) = E (3) ∩ C (3) = (0, 0). Finally the tangent cone, which is given by h 2 + hb = h(h + b), is the union of two different lines. So P2(3) is an ordinary double point of C (3) . Thus, two branches of C 2k intersect at P2 . If both of them belong to R (or S), then i(R, S; P2 ) = 0. We consider the case that two branches are in different components. The multiplicities in the points P2 , P2(1) , P2(2) , P2(3) must all be 1. Analogous to the case p = 2 it follows i(R, S; P2 ) = 1 · 1 + 1 · 1 + 1 · 1 + 1 · 1 = 4. The remaining point P3 = (0, 0) in the affine patch U2 = {(h : a : 1) ∈ P2 (Fq )} has a blow up sequence similar to the point P1 . Now we can prove the irreducibility of C q . Lemma 2.6 Let p be a prime ∈ / {37, 431} and q = p k . Then the curve C q is irreducible and has degree 6. , ∂ F(1,a,b) ) has dimension 0 (see proof of Lemma 2.4). Proof V (F(1, a, b), ∂ F(1,a,b) ∂a ∂b Hence, F(h, a, b) is square free, and therefore, C q has degree 6. Assume that C q is reducible, i.e., C q = R ∪ S. The singular points {P1 , P2 , P3 } of C q are possible intersection points of R and S. In Lemma 2.5 we proved: 3 1. i(R, S; P j ) ∈ {0, 1, 2, 3, 4}, if q = 2k and 3j=1 k 2. j=1 i(R, S; P j ) ∈ {0, 4}, if q = 2 . This contradicts Bezout’s theorem, as noted at the beginning of this subsection.
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2.2.2 Estimate of |C q | Remark 2.7 |V37 (J )| = 38 and |V431 (J )| = 440 (computed by MAGMA). Lemma 2.8 |Vq (J )| = ∅ if q = 8. Proof The arithmetic genus of C q is pa = (d−1)(d−2) = 5·4 2 2 = 10 by [8, I.7.2], where d denotes degree of C q . By the Hasse–Weil estimate (Theorem 2.3) we have √ |C q | ≥ (q + 1) − 20 p. Hence, |C q | > 9 if q ≥ 416. It follows from Lemma 2.2 that |Vq (J )| = ∅ if q ≥ 416. For 5 ≤ q ≤ 415 we computed |Vq (J )| by MAGMA and obtained |Vq (J )| = ∅ if q = 8. Remark 2.9 For the case q = 8 we check by MAGMA directly that there are (x, y) ∈ PSL(2, 8) which satisfy v1 (x, y) = 1 and v1 (x, y) = v2 (x, y). 2.3 Case Sz(2n ) m+1
Let n := 2m + 1 and θ : F2n −→ F2n , defined by θ (a) = a 2 . As shown in [14, p. 133] and [10, XI.3], the Suzuki group Sz(2n ) is generated by S(a, b), M(λ), T | a, b, λ ∈ F2n , λ = 0 ≤ G L(4, F2n ), where ⎛ ⎜ ⎜ ⎜ S(a, b) = ⎜ ⎜ ⎜ ⎝ ⎛ ⎜ ⎜ ⎜ M(λ) = ⎜ ⎜ ⎜ ⎝
1
0
a
1
0 0
⎞
⎟ 0 0⎟ ⎟ ⎟, ⎟ aθ (a) + b θ (a) 1 0 ⎟ ⎠ 2 a θ (a) + ab + θ (b) b a 1 λ1+2
m
0 m
0
0
0
λ2
0
0
λ−2
0
0
0
0
⎛
⎞
0 0 0 1
⎟ ⎜ ⎟ ⎜0 0 1 0⎟ ⎟ ⎟ ⎜ ⎟ ⎟. ⎟ and T = ⎜ ⎟ ⎜ ⎟ ⎟ ⎜ ⎟ 0 1 0 0 0 ⎠ ⎝ ⎠ m −1−2 1 0 0 0 λ 0
m
For the proof of Theorem 1.2 we make the ansatz ⎛
0 0
0
1
⎞
⎜ ⎟ ⎜0 0 1 ⎟ a ⎜ ⎟ ⎟ and x := S(a, b)T = ⎜ ⎜ ⎟ ⎜ 0 1 θ (a) ⎟ aθ (a) + b ⎝ ⎠ 2 1 a b a θ (a) + ab + θ (b)
123
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⎛ ⎜ ⎜ ⎜ y := S(c, d) = ⎜ ⎜ ⎜ ⎝
395
1
0
c
1
0 0
⎞
⎟ 0 0⎟ ⎟ ⎟ ∈ F2 [a, b, c, d]. ⎟ cθ (c) + d θ (c) 1 0 ⎟ ⎠ 2 c θ (c) + cd + θ (d) d c 1
Remark 2.10 v1 (x, y) = 1 holds if and only if a = b = c = d = 0. Because x and y depend on n we replace θ (a), θ (b), θ (c), θ (d) by new variables a0 , b0 , c0 , d0 (as in [2]). Hence we have ⎛
0 0
0
1
1
0 0 0
⎞
⎟ ⎜ ⎟ ⎜0 0 1 a ⎟ ⎜ ⎟ and ⎜ x(a, b, a0 , b0 ) := ⎜ ⎟ ⎟ ⎜ 0 1 a0 aa0 + b ⎠ ⎝ 1 a b a 2 a0 + ab + b0 ⎛ ⎜ ⎜ ⎜ y(c, d, c0 , d0 ) := ⎜ ⎜ ⎜ ⎝
c cc0 + d c2 c0 + cd + d0
⎞
⎟ 1 0 0⎟ ⎟ ⎟ ⎟ c0 1 0 ⎟ ⎠ d c 1
as matrices over R := F2 [a, a0 , b, b0 , c, c0 , d, d0 ]. Let J be the ideal in R corresponding to the condition v1 (x, y) = v2 (x, y) in 8 R. Any point in V (J ) ⊆ F2 satisfies v1 (x, y) = v2 (x, y) but is not necessarily 8 defined over F2n . To find such points (for any n) we use automorphism: α: F2 −→ 8 F2 , α(a, a0 , b, b0 , c, c0 , d, d0 ) = (a0 , a 2 , b0 , b2 , c0 , c2 , d0 , d 2 ). If (a, a0 , b, b0 , c, c0 , 8 d, d0 ) ∈ F2 is a fixed point of α n , then (a, a0 , b, b0 , c, c0 , d, d0 ) ∈ F82n (see [2]). Hence the problem is reduced to the case to show that α n has fixed points on V (J ) for all odd n ≥ 3. To show this, we use the following Theorem, which is a special case of 8 Lefschetz’s fixed point formula. For a subset U ⊆ F2 let
8 Fix(U, n) := v ∈ F2 | α n (v) = v denote the set of fixed points of α n on U and Fix(U, n) the projective closure of Fix(U, n) in P8 (F2 ). Theorem 2.11 [2, 3.6] Let n = 2m + 1 and U a smooth, α-invariant, irreducible 8 subset of F2 which has dimension 2 and |Fix(U, n)| = | Fix(U, n)|. Then | Fix(U, n)| ≥ 2n − b1 (U )23n/4 − b2 (U )2n/2 ,
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where bi (U ) = dim Heti (U, Q ) are the -adic Betti numbers ( = 2). Thus we have to find a smooth, α-invariant, irreducible subset of V (J ) and estimate the Betti numbers. To start, we will remove some components of V (J ) and call the new set V (J0 ). Then we will search for a polynomial f (a, a0 ) ∈ F2 [a, a0 ] such that 8 8 3 V (J0 ) \ V ( f ) ⊆ F2 is smooth, α-invariant, irreducible and the projection F2 → F2 onto the (a, a0 , b)-three space is an isomorphism. The last condition will enable us to get an easy estimate on the Betti numbers. We remove components of V (J ) in the hyperplanes {a = 0} and {a0 = 0}. Ja := (J ∩ at − 1) ∩ R ⊆ R,
J0 := (Ja ∩ a0 t − 1) ∩ R ⊆ R
Here at − 1 and a0 t − 1 are ideals over R[t]. The corresponding computer algebra programs of this and further computations can be found in [13]. Remark 2.12 An easy SINGULAR computation shows that the new set V (J0 ) is α-invariant. Namely, define π : R −→ R, by π(a) = a0 , π(a0 ) = a 2 , . . . , π(d) = d0 , π(d0 ) = d 2 , and check π(J0 ) ⊆ J0 . A Groebner basis of J0 with respect to the graduate reverse lexicographical ordering dp [7, 1.2] can be computed by SINGULAR in few minutes. However, we want to project V (J0 ) into an affine space of smaller dimension. The projection can be examined more easily if one has a Groebner basis with respect to the lexicographical ordering lp, but the computation of this basis exceeded the limits of the current computers. Nevertheless, we computed generators for J0 which enable us to define 3 and study a projection into F2 . The idea is to use several product term orderings in SINGULAR to obtain especially nice elements of the Groebner basis. At first, we computed Groebner basis of J0 where d0 has the lexicographical ordering lp and the remaining variables d, c0 , c, b0 , b, a0 , a graduate reverse lexicographical ordering dp. This gives us a basis of J0 , which contains only one polynomial involving d0 and this is of type d0 + p0 (a, a0 , b, b0 , c, c0 , d). This means that the projection of V (J0 ) along the d0 -axis is biregular; the inverses being (a, a0 , b, b0 , c, c0 , d) → (a, a0 , b, b0 , c, c0 , d, p0 (a, a0 , b, b0 , c, c0 , d)). We remove the polynomial d0 + p0 (a, a0 , b, b0 , c, c0 , d) from J0 and call the resulting ideal J0d0 . Next, we can deal with the variables d, c0 in the same way resulting in polynomials of type d + p1 (a, a0 , b, b0 , c, c0 ) and c0 + p2 (a, a0 , b, b0 , c) and respective ideals J0d0 ,d and J0d0 ,d,c0 . Now our luck runs out, we cannot treat any of the remaining variables in the same way. We find further simple polynomials in the ideal J0d0 ,d,c0 by eliminating b0 from it, and then computing a Groebner basis with respect to the product term ordering where c is of the highest order. In this basis we find 131 polynomials linear in c, i.e., c · γi (a, a0 , b) + p3,i (a, a0 , b). Reversing the roles of b0 and c in the proceeding step we obtain 125 polynomials linear in b0 , i.e., b0 · β j (a, a0 , b) + p4, j (a, a0 , b). Finally, eliminating b0 and c from J0d0 ,d,c0 we get an ideal in (a, a0 , b) that is generated by one polynomial which we call h. In the whole we can think of J0 as being generated in the following way:
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J0 = d0 d c0 c · γ1 (a, a0 , b) .. .
+ + + +
397
p0 (a, a0 , b, b0 , c, c0 , d), p1 (a, a0 , b, b0 , c, c0 ), p2 (a, a0 , b, b0 , c), p3,1 (a, a0 , b), .. .
c · γ131 (a, a0 , b) + p3,131 (a, a0 , b), b0 · β1 (a, a0 , b) + p4,1 (a, a0 , b), .. .. . . b0 · β125 (a, a0 , b) + p4,125 (a, a0 , b), h(a, a0 , b), polynomials over F2 [a, a0 , b, b0 , c] which contain ck for k ≥ 2 polynomials over F2 [a, a0 , b, b0 ] which contain b0k for k ≥ 2.
Note that this immediately implies that the projection of V (J0 ) onto V (h) in the (a, a0 , b)-space is a birational map with a birational morphism outside V (γ1 , . . . , γ131 ) ∪ V (β1 , . . . , β125 ). Let E := γ1 , . . . , γ131 ∩ β1 , . . . , β125 ∩
∂h(a, a0 , b) ∂h(a, a0 , b) ∂h(a, a0 , b) , , ∂a ∂a0 ∂b
be the intersection of ideals in F2 [a, a0 , b]. We compute the generator f˜ of the radical ideal of E ∩ F2 [a, a0 ], i.e., Rad(E ∩ F2 [a, a0 ]) = f˜, and set f (a, a0 ) := f˜ · π( f˜), where π(a) = a0 , and π(a0 ) = a 2 . Lemma 2.13 The set U := V (J0 ) \ V ( f (a, a0 )) is non-empty, smooth, α-invariant, irreducible, and isomorphic to W := V (h(a, a0 , b)) \ V ( f (a, a0 )). / J0 . Proof (1) V (J0 ) \ V ( f (a, a0 )) is non-empty since f ∈ (2) The set of singular points of V (J0 ) is contained in V (E) ⊆ V ( f (a, a0 )), because the Jacobian of ⎛
1 ⎜0 ⎜ ⎜0 ⎜ ⎜0 ⎜ ⎜. ⎜ .. ⎜ ⎜ J0 Jac(J0 ) = ⎜ 0 ⎜0 ⎜ ⎜. ⎜. ⎜. ⎜ ⎜0 ⎜ ⎝0
⎞ 1 0 1 0 0 x1 (a, a0 , b) .. .. . .
0 0 x131 (a, a0 , b) 0 0 0 .. . 0 0 0 0 0 0
y1 (a, a0 , b) .. . y125 (a, a0 , b) ∂h(a,a0 ,b) 0 ∂a ...
∂h(a,a0 ,b) ∂a0
⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ∂h(a,a0 ,b) ⎟ ⎠ ∂b
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has maximal rank 6 for points in V (J0 ) \ V (E). Hence, V (J0 ) \ V ( f ) is smooth. (3) Therefore, we showed in Remark 2.12 that V (J0 ) is α-invariant. Therefore, for 8 the α-invariance of V (J0 ) \ V ( f ) it suffices to show that F2 \ V ( f ) is α-invariant. We / V ( f ) then α(v) ∈ / V ( f ). show that if v = (v1 , . . . , v8 ) ∈ Assume that α(v) ∈ V ( f ). Since f = f˜ · π( f˜), we have f˜(α(v)) = 0 or π( f˜(α(v))) = 0. If f˜(α(v)) = 0 then by f˜(α(v)) = π( f˜(v)) = 0 we have v ∈ V ( f ) and thereby a contradiction. If π( f˜(α(v))) = 0 then we get π( f˜(α(v))) = f˜ 8 (α 2 (v)) = f˜(v12 , . . . , v82 ) = 0. Because the characteristic of F2 is 2, it follows that / V ( f ). f˜2 (v) = 0 and hence f (v) = 0 contradicting v ∈ (4) V (J0 ) \ V ( f ) and V (h(a, a0 , b)) \ V ( f ) are isomorphic. One can see this comparing the new basis of J0 and the construction of V (E), respectively, V ( f ). (5) Because of the isomorphism, V (J0 ) \ V ( f ) is seen to be irreducible once one shows that the polynomial h(a, a0 , b) is irreducible. However, we can also use a part of the proof of [2]. At first eliminate in J0 variables d0 , d, c0 , b0 , a0 . This gives us a polynomial j (a, b, c) and a rational projection V (J0 ) → V ( j (a, b, c)) (for details see [13]). j (a, b, c) is a polynomial of degree 35 over F2 [a, b, c]. Now we show that j (a, b, c) has no factors in F2 [a, c] (the same proof as in [2]). Then we consider j (1, c, xc )/ c2 = x 12 + c2 x 10 + (c7 + c6 + c5 + c3 + c)x 8 + (c9 + c4 + c3 )x 6 + (c11 + c10 + c9 + c8 + c6 + c4 + c2 )x 4 + c6 x 2 + (c10 + c8 ). By [2, 3.3] this polynomial is irreducible over F2 [a, b, c] and c2 can not be a factor of j (a, b, c). Therefore, V (J0 ) is irreducible. Now we want to estimate the -adic Betti numbers b1 and b2 of U W . Lemma 2.14 b1 (W ) ≤ 4482, where b1 (W ) = dim Het1 (W, Q ). 3
Proof Let H be a general hyperplane of F2 . By Lefschetz hyperplane section theorem the map Het1 (W, Q p ) → Het1 (W ∩ H, Q p ) is injective [12] thus b1 (W ) ≤ b1 (W ∩ H ). Let W ∩ H be the projective closure of W ∩ H in H P2 (F2 ). Since the surface W is irreducible by Lemma 2.13, W ∩ H is an irreducible curve in H P2 (F2 ). By [6, 7.4] we have b1 (W ∩ H ) ≤ (d − 1)(d − 2), where d denotes degree of W ∩ H ⊆ P2 (F2 ). It follows from the construction of W that W ∩ H = W ∩ H = V (h) \ V ( f ) ∩ H = V (h) ∩ H . Since W ∩ H is irreducible Bezout’s theorem [8, I.7.7] gives us: deg W ∩ H ≤ deg V (h) · deg H = 35 · 1 = 35. Therefore, b1 (W ∩ H ) ≤ (35 − 1)(35 − 2) = 1122. However, we need an estimate for b1 (W ∩ H ). The sequence Het1 (W ∩ H , Ql ) → Het1 (W ∩ H, Ql ) → Het0 (W ∩ H \ (W ∩ H ), Ql ) is exact, thus b1 (W ∩ H ) ≤ b1 (W ∩ H ) + dim Het0 (W ∩ H \ (W ∩ H ), Ql ).
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The set W ∩ H \ W ∩ H consists of finitely many points, in particular dim Het0 (W ∩ H \ W ∩ H, Ql ) ≤ deg W ∩ H \ W ∩ H. It follows from the construction of W that W ∩ H \ W ∩ H = (W ∩ H ) ∩ (V ( f ) ∪ (H \ H )). By Bezout’s theorem we have: deg(W ∩ H \ W ∩ H ) ≤ deg W ∩ H · deg(V ( f ) ∪ (H \ H )) ≤ 35 · (95 + 1) = 3360. Altogether: b1 (W ) ≤ b1 (W ∩ H ) ≤ b1 (W ∩ H ) + deg(W ∩ H \ W ∩ H ) ≤ 1122 + 3360 = 4482. Lemma 2.15 b2 (W ) ≤ 10828656 ≤ 224 , where b2 (W ) = dim Het2 (W, Q ). Proof We start by showing that the Euler characteristic of W can be estimated by: |χ (W )| ≤ 10824174. By definition we have: V (h) = (V (h) \ V ( f )) ∪ (V (h) ∩ V ( f )) = W ∪ V (h, f ). Since the sets W and V (h, f ) are disjoint, we get χ (W ) = χ (V (h)) − χ (V (h, f )). For the estimates we use the following theorem (as in [2]). Theorem 2.16 [1,11] If an affine variety V is defined in A N by r polynomial equations of degree ≤ d, then |χ (V )| ≤ 2r D N ,r (1, 1 + d, . . . , 1 + d), r +1
W is the homogeneous form of degree N in where D N ,r (x0 , . . . , xr ) = |W |=N X x0 , . . . , xr all of whose coefficients equal 1. In our case we need only: D3,1 (x0 , x1 ) = x03 +x02 x1 +x0 x12 +x13 and D3,2 (x0 , x1 , x2 ) = x03 + x02 x1 + x02 x2 + x0 x12 + x0 x1 x2 + x0 x22 + x13 + x12 x2 + x1 x22 + x23 . We obtain |χ (V (h))| ≤ 2· D3,1 (1, 35+1) = 2·(1+36+362 +363 ) = 95978 and |χ (V (h, f ))| ≤ 22 · D3,2 (1, 95 + 1, 95 + 1) = 4 · (1 + 2 · 96 + 3 · 962 + 3 · 963 ) = 10728196. Thus |χ (W )| = |χ (V (h))| + |χ (V (h, f ))| ≤ 95978 + 10728196 = 10824174. If χ (W ) ≤ 0, then 1−b1 (W )+b2 (W ) ≤ 0, hence b2 (W ) < b1 (W ) ≤ 4482 < 212 . Otherwise if χ (W ) > 0, we have b2 (W ) = −1 + χ (W ) + b1 (W ) ≤ 10824174 + 4482 = 10828656 ≤ 224 . Now we can prove that α n has fixed points on V (J ) for all odd n ≥ 3. Theorem 2.17 Fix(U, n) = ∅ for all odd n ≥ 3. Let n = 2m + 1, we want to apply Theorem 2.11 to estimate | Fix(U, n)| for large n. The set U is by Lemma 2.13 non-empty, smooth, α-invariant and irreducible, in order to apply the Theorem 2.11 it remains to show: |Fix(U, n)| = | Fix(U, n)|. We denote by H∞ = {(0 : a : b : c : d : a0 : b0 : c0 : d0 )} ⊆ P8 (F2 ) the hyperplane at infinity of P8 (F2 ). Then U \ U = (V (J0 ) ∩ H∞ ) ∪ (V (J0 ) ∩ V ( f )) and hence Fix(U, n) \ Fix(U, n) = (Fix(V (J0 ), n) ∩ H∞ ) ∪ (Fix(V (J0 ), n) ∩ V ( f )). If Fix(V (J0 ), n) ∩ V ( f ) is non-empty, we have a fixed point of α n over V (J0 ) and the Theorem 1.2 is proved. Therefore, we may assume that Fix(V (J0 ), n) ∩ V ( f ) = ∅.
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We will show that Fix(V (J0 ), n) ∩ H∞ = ∅. Let (t, a, a0 , b, b0 , c, c0 , d, d0 ) ∈ m m+1 m n Fix(V (J0 ), n) ∩ H∞ , then α 2m+1 (a) = a02 = (a 2 )2 = a 2 , hence the n
a-coordinate is fixed if a = a 2 . The homogeneous ideal which describes Fix(P8 (F2 ), n) n n contains the polynomial at 2 −1 = a 2 . Since t = 0 in H∞ we have a = 0, and therefore, a0 = 0. Analogous we show that b = c = d = 0. Now we can apply Theorem 2.11 | Fix(U, n)| ≥ 2n − 4482 · 23n/4 − 10828656 · 2n/2 , and obtain | Fix(U, n)| > 1 if n > 50. For 3 ≤ n ≤ 50 we can find examples of fixed points of α n by MAGMA (it suffices to consider primes n). It is not possible to check all points of F82n because there are too many of them, for example F8247 has approximately 1057 points. Examples are found by the following method: Let p = 2m + 1 be a prime, 3 ≤ p ≤ 50 and t be the polynomial variable of the finite field F2 p = F2 [t]/(g(t)). We choose m+1 a ∈ {0, 1, t, t + 1, t 2 , t 2 + 1, t 2 + t, t 2 + t + 1, t 3 , . . . } and compute a0 := a 2 . Further, the coordinate b has to be a root of the polynomial h(a, a0 , b), which has degree 12 with respect to b. We use the Variety( ); algorithm in MAGMA to compute b. For the further coordinates we use the polynomials of the new basis of J0 . In this way examples for all p are found, for details see [13]. 2.4 Remarks (1)
(2)
The methods of the proof of Suzuki case (to find a new basis of the ideal) can also be used to prove the Suzuki case in [2]. One has to eliminate the variables in order d0 , d, c0 , c, b0 and get exactly the same polynomial h(a, a0 , b) as in our proof. Note that the original ideals J in Suzuki case in [2] and here are not the same. Consider a sequence v1 (x, y) = f, vn (x, y) = [vn−1 (x, y)g , vn−1 (x, y)h ] for some words f, g, h. Then the case PSL(2, Fq ) can most likely be proven, if V (I ) is a curve over Q, where I is the ideal corresponding to v1 (x, y) = v2 (x, y). If f, g, and h are words of the length at most 3, then there are approximately 300 sequences (see [13]) where this is the case.
Acknowledgments This article is part of the author’s Ph.D thesis, which was supervised by Fritz Grunewald. The author thanks F. Grunewald and J. Piontkowski for useful comments and advise.
References 1. Adolphson, A., Sperber, S.: On the degree of the L-function associated with an exponential sum. Compos. Math. 68, 125–159 (1988) 2. Bandman, T., Greuel, G.-M., Grunewald, F., Kunyavskii, B., Pfister, G., Plotkin, E.: Engel-like identities characterising finite solvable groups. Compos. Math. 142, 734–764 (2006) 3. Brandl, R., Wilson, J.-S.: Characterisation of finite soluble groups by laws in a small number of variables. J. Algebra 116, 334–341 (1988)
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4. Bray, J.-N., Wilson, J.-S., Wilson, R.-A.: A characterisation of finite soluble groups by laws in two variables. Bull. Lond. Math. Soc. 37, 179–186 (2005) 5. Fulton, W.: Intersection Theory. Springer, Berlin (1998) 6. Ghorpade, S.-R., Lachlaud, G.: Etale cohomology, Lefschetz theorems and number of points of singular varieties over finite fields. Mosc. Math. J. 2, 589–631 (2002) 7. Greuel, G.-M., Pfister, G.: A SINGULAR Introduction to Commutative Algebra. Springer, Berlin (2002) 8. Hartshorne, R.: Algebraic Geometry. Springer, New York (1977) 9. Huppert, B.: Endliche Gruppen I. Springer, Berlin (1967) 10. Huppert, B., Blackburn, N.: Finite Groups III. Springer, Berlin (1982) 11. Katz, N.-M.: Sums of Betti numbers in arbitrary characteristics. Finite Fields Appl. 7, 29–44 (2001) 12. Milne, J.-S.: Etale Cohomology. Princeton University Press, Princeton (1980) 13. Ribnere, E.: Engelbedingungen für nilpotente und auflösbare Gruppen Dissertation, Universität Düsseldorf, Düsseldorf (2007) 14. Suzuki, M.: On a class of doubly transitive groups. Ann. Math. 75(2), 105–145 (1962) 15. Thompson, J.: Non-solvable finite groups all of whose subgroups are solvable. Bull. Am. Math. Soc. 74, 383–437 (1968) 16. Zorn, M.: Nilpotency of finite groups. Bull. Am. Math. Soc. 42, 485–486 (1936)
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