Analyzing finitely presented groups by constructing representations

W. Plesken  B. Souvignier

1 Introduction

One idea how to prove that a finitely presented group G is infinite is to construct suitable homomorphisms into infinite matrix groups. In [HoP 92] this is done by starting with a finite image H of G and solving linear equations to check whether the epimorphism onto H can be lifted to a representation whose image is an extension of a ZZ-lattice by H, thus exhibiting an infinite abelian section of G in case it exists near the top of the group. The variation of the idea presented here seems to be suitable for instance for some groups where the kernel of the epimorphism onto H has a pro-p-completion of bounded width. The price to be paid is that algebraic rather than linear equations have to be solved. These are obtained from evaluating the relations on matrices with indeterminates as entries. This system of equations can be simplified by applying some representation theory, but it often remains too complicated for direct methods e.g. Gröbner base calculations or elimination of variables by taking resultants. An alternative to the direct methods is to find a modular solution and to lift it to a solution over a p-adic field. Knowing a finite epimorphic image H of G, H and its modular representation theory might give us a hint which modular representation of H one should try to lift to a p-adic representation of G. A multi-dimensional version of Hensel’s lemma often assures that it is sufficient to lift a solution only a few steps to conclude that it can be lifted to a solution over a p-adic field. Note, although the initial equations are algebraic, each step of the lifting process is just a problem of solving linear equations over a finite field. In general it is not clear that the lifting process leads to entries which are algebraic over I Q even in the situation where each lift is equivalent to a representation whose entries are algebraic over I Q. To get a representation defined over a number field K (with degree [K : I Q] as small as possible), therefore involves two steps: To pass from the lifted representation to one with p-adic entries which are algebraic over I Q and to identify the new entries as algebraic numbers. In the examples of this paper the first step turned out to be superfluous, since the lifted representation had already entries with rational minimal polynomials of small degree, see however [HPS 94] for a more ill-behaved example. The second step is discussed in Section 5. Sometimes the lift does not work in characteristic 0, however is possible in the ring IF_{p}[[x]] of Laurent series over IF_{p}. An even simpler method than the one for number fields yields a representation over the field of rational functions over IF_{p}. In both cases one obtains infinitely many finite images of G by reducing modulo prime ideals of K resp. IF_{p}[[x]]. These images are often simple and thus the method often provides infinitely many simple factor groups by starting with a single one at the beginning of the computation. We give some examples to illustrate our methods. The groups in Examples 3 and 5 were already treated in [Wes 85] whereas infiniteness of the groups in the Examples 1, 2 and 4 was open. In [HPS 94] a representation of the group (2, 3, 7; 11) of degree 7 is constructed by the method developed here. Most computations were performed in MAPLE. However, it is clear that more adjusted programs are highly desirable. What is needed are: polynomial arithmetic over algebraic number fields and finite fields including Gröbner bases and resultants, p-adic arithmetic, matrix arithmetic over these domains, furthermore LLL-reduction and Gaussian elimination for very large numbers and a program for computing maximal orders and class numbers of algebraic number fields.

We thank D. Holt for providing us with the examples, A. Cohen for sharing his experience with Gröbner bases and resultants for solving algebraic equations with us, and M. Pohst for pointing out the reference [dWe 89] to us.

2 The direct method

Throughout this paper let G be a finitely presented group with presentation

G := ⟨g_{1}, …, g_{n}|w_{1}(g_{1}, …, g_{n})^{r_{1}}, …, w_{m}(g_{1}, …, g_{n})^{r_{m}}⟩.

We want to find a representation of G of some degree d. A naive approach would be to map the generators g_{k} on matrices γ_{k} := (x_{ij}^{(k)}) in indeterminates x_{ij}^{(k)} and to take the entries of w_{i}(γ_{1}, …, γ_{n})^{r_{i}} −I_{d} as equations. This leads to a system of m · d^{2} equations in n · d^{2} indeterminates with a possibly very high degree. It will almost never be possible to solve such a system of equations by direct methods, but examples are known in the literature where a qualitative analysis of the equations leads already to results on G.

The following two remarks can often be used to obtain a simpler system of algebraic equations:

1) If the relators w_{1}, …, w_{t} only involve the generators g_{1}, …, g_{s} and this restricted presentation defines a finite subgroup U of G, then a representation of G must restrict to a representation of U. Therefore representation theory of finite groups gives explicit candidates for the images γ_{1}, …, γ_{s} of g_{1}, …, g_{s}. This reduces the number of equations, their degrees and the number of indeterminates.

2) One has only finitely many possibilities for the characteristic polynomial of w_{i}(γ_{1}, …, γ_{n}) and therefore can take the coefficients of the characteristic polynomials of these matrices as equations. This reduces the number of equations and often their degrees.

By these two remarks one obtains a system of (m − t) · d equations in (n − s) · d^{2} indeterminates. Clearly each equation which is linear in one of the indeterminates can be used to reduce both the numbers of equations and of indeterminates by one. Especially for low matrix degrees d such a system of algebraic equations can sometimes be solved by one of the following direct methods:

1) The resultant method The classical method to solve a system of equations is to eliminate indeterminates by taking resultants and then solve the system by backward substitution as described in [GCL 92], Chapter 9. Choosing different paths of elimination, one often succeeds in obtaining some equations in only one indeterminate. Let f_{1}, …, f_{l} be such equations in the same indeterminate. Then these equations can be replaced by their greatest common divisor f. If f = 1 no solution in characteristic 0 is possible. In this case some of the resultants r_{ij} := res(f_{i}, f_{j}) are non-zero integers and the prime divisors of the greatest common divisor R of the r_{ij} are the candidates for prime numbers, where a modular solution exists, since taking resultants works over ZZ. This method was already used in [HoP 92], Section 5, to find prime numbers p such that P SL_{2}(p) is an image of a finitely presented group.

2) The Gröbner base method A second method is to compute a Gröbner basis (cf. [Buc 65]) for the ideal generated by the equations in the ring ZZ[x_{ij}^{(k)}]. There exists a solution in characteristic 0 if and only if the Gröbner basis does not collapse to (1), i.e. the ideal is not the whole ring. If the ideal is zero-dimensional (i.e. the residue class algebra is finite dimensional) one can choose a suitable basis for the residue class algebra and express the indeterminates in this basis by means of the unique normal form with respect to the Gröbner basis. If the ideal has a higher dimension one will usually first specialize indeterminates until the ideal becomes zero-dimensional. Namely, to prove infiniteness one does not need the most general solution, but one (for instance with algebraic entries) with an infinite image.

In practice it turned out to be fruitful to use a combination of these two methods. On the one hand one obtains by the Gröbner base method nice expressions for the entries if one uses the powers of a trace of some matrix as basis for the residue class algebra. On the other hand the resultant method often yields minimal polynomials for the entries of the matrices. Adding these to the equations makes it much easier to calculate a Gröbner basis.

The following two examples come from a family of presentations denoted by (m, n, p; q) which is an abbreviation for ⟨a, b|a^{m}, b^{n}, (ab)^{p}, [a, b]^{q}⟩.

Example 1: Let G = (3, 4, 8; 2), then the commutator subgroup G′ has index 4 in G and a presentation ⟨a, b, c, d|a^{3}, b^{3}, c^{3}, d^{3}, (a^{−1}d)^{2}, (b^{−1}a)^{2}, (c^{−1}b)^{2}, (d^{−1}c)^{2}, (abcd)^{2}⟩. We try to construct a projective representation of degree 2 of G′, i.e. the images α, β, ζ, δ of a, b, c, d satisfy the above relations only modulo scalar matrices. Since G′ is perfect, one ought to assume that α, β, ζ, δ have determinant equal to 1.

Clearly ⟨a, b|a^{3}, b^{3}, (b^{−1}a)^{2}⟩ ≅ Alt(4), thus we start by mapping a on α := ( \begin{matrix}-1 & 1 \\ -1 & 0 \end{matrix} )

and b on β := ( \begin{matrix}-1 & -i \\ -i & 0 \end{matrix} ), where i^{2} = −1. Let ζ := ( \begin{matrix}x_{1} & x_{2} \\ y_{1} & y_{2} \end{matrix} ) and δ := ( \begin{matrix}x_{3} & x_{4} \\ y_{3} & y_{4} \end{matrix} )

be the images of c and d, then tr(ζ) = tr(δ) = −1 since c and d have order 3. Hence y_{2} = −1 − x_{1} and y_{4} = −1 − x_{3}. Furthermore from tr(ζ^{−1}β) = tr(α^{−1}δ) = 0 one gets y_{1} = i + ix_{1} − x_{2} and y_{3} = 1 + x_{3} + x_{4}. Thus one obtains the following system of nonlinear equations in x_{1}, x_{2}, x_{3}, x_{4}: f_{1} : det(ζ) − 1 = 0, f_{2} : det(δ) − 1 = 0, f_{3} : tr(δ^{−1}ζ) = 0, f_{4} : tr(αβζδ) = 0 By the resultant method over I Q[i] ≅ I Q[x]/(x^{2} +1) one succeeds in finding quadratic relations m_{j} = m_{j}(x_{j}) for each x_{j}, which as polynomials in I Q[i][x_{j}] are irreducible. Ideally the Gröbner base method over I Q[i] now yields that I Q[i][x_{1}, x_{2}, x_{3}, x_{4}]/(f_{1}, f_{2}, f_{3}, f_{4}, m_{1}, m_{2}, m_{3}, m_{4}) ≅ I Q[i][x]/(x^{2} + ix + i). Since no I Q[i]-Gröbner base implementation is available to us, Gröbner base methods applied to I Q[y, x_{1}, x_{2}, x_{3}, x_{4}]/(y^{2} + 1, f̃_{1}, …, m̃_{4}) yield I Q[x]/(x^{4} + x^{2} + 2x + 1), where f̃_{j} etc. is obtained from f_{j} by replacing i by y. Now tr(αζβ) generates the field K := I Q[θ], where θ is a root of x^{4} + x^{2} + 2x + 1. With the help of KANT (cf. [Poh 93]) one checks that K has discriminant 2^{4} · 17, classnumber 1, no real embedding into I C, maximal order ZZ[θ] and Galois group D_{8}. The last fact proves that the matrix αζβ has infinite order, because elements of finite order have traces in cyclotomic number field, which have abelian Galois groups. The entries of the matrices can be expressed as: i = θ^{3} − θ^{2} + θ + 1, x_{1} = −\frac{1}{2}θ^{3} − 2θ − \frac{1}{2}, x_{2} = \frac{3}{2}θ^{3} − \frac{1}{2}θ^{2} + \frac{5}{2}θ + 1, x_{3} = 2θ^{3} − \frac{3}{2}θ^{2} + \frac{7}{2}θ + \frac{5}{2}, x_{4} = −\frac{3}{2}θ^{3} − 2θ − \frac{7}{2}. The fractions can be eliminated by conjugating the representation with the matrix

( \begin{matrix}1+θ^{3} & -θ+θ^{2}-θ^{3} \\ 0 & 2 \end{matrix} ), which gives the following representation over ZZ[θ]:

a ↦ ( \begin{matrix}2θ-θ^{2}+θ^{3} & -1-2θ+θ^{2}-θ^{3} \\ -1+θ-θ^{2} & -1-2θ+θ^{2}-θ^{3} \end{matrix} ), b ↦ ( \begin{matrix}-3-θ-θ^{3} & -2θ+θ^{2}-θ^{3} \\ -3-3θ+θ^{2}-2θ^{3} & 2+θ+θ^{3} \end{matrix} ),

c ↦ ( \begin{matrix}-1-4θ+2θ^{2}-2θ^{3} & 4+6θ-3θ^{2}+4θ^{3} \\ 3+θ^{2} & 4θ-2θ^{2}+2θ^{3} \end{matrix} ), d ↦ ( \begin{matrix}6+7θ-3θ^{2}+5θ^{3} & -9-5θ+2θ^{2}-5θ^{3} \\ 2+5θ-2θ^{2}+3θ^{3} & -7-7θ+3θ^{2}-5θ^{3} \end{matrix} ).

¿From this representation one obtains homomorphisms into P SL_{2}(q) where q = p^{f} such that x^{4} +x^{2} +2x+1 has a zero in IF_{q}. For example one obtains homomorphisms into P SL_{2}(p) for p = 5, 17, 29, 37, 41, 53, 61, 73, 97, …, which one can easily prove to be epimorphisms. Of course one can also reduce modulo arbitrarily high powers of prime ideals and get p-groups of arbitrarily large class underneath the P SL_{2}(p). Cf. [HoP 93], [HoP 89], Section 2.3.3 for the philosophical background, which also leads to the more refined methods in Sections 3 and 4 of this paper. If one prefers a linear rather than a projective representation one can take the action on the symmetric tensor square of the 2-dimensional module.

Example 2: Let G = (3, 4, 14; 2), then the commutator subgroup G′ has index 2 in G and a presentation ⟨a, b, c|a^{3}, b^{3}, c^{2}, (a^{−1}b)^{2}, (a^{−1}cbc)^{2}, (abc)^{7}⟩. Again we try to construct a projective representation of degree 2.

As in the above example we map a on α := ( \begin{matrix}-1 & 1 \\ -1 & 0 \end{matrix} ) and b on β := ( \begin{matrix}-1 & -i \\ -i & 0 \end{matrix} ),

where i^{2} = −1. Let ζ := ( \begin{matrix}x_{1} & x_{2} \\ y_{1} & y_{2} \end{matrix} ) be the image of c, then tr(ζ) = 0, hence

y_{2} = −x_{1} and tr(αβζ) = ξ_{7} + ξ_{7}^{−1} where ξ_{7} is a primitive 7th root of unity. One thus ends up with only two equations: f_{1} : det(ζ) = 1 and f_{2} : tr(α^{−1}ζβζ) = 0. By the resultant method over I Q[i, ξ_{7} + ξ_{7}^{−1}] one obtains relations m_{1}, m_{2} of degree 4 for x_{1}, x_{2}. With Gröbner base methods one computes that the residue class algebra I Q[i, ξ_{7} + ξ_{7}^{−1}][x_{1}, x_{2}]/(f_{1}, f_{2}, m_{1}, m_{2}) is isomorphic to the field I Q[i][x]/(m_{θ}), where m_{θ} := x^{12} − x^{11} − 6x^{10} + 7x^{9} + 15x^{8} − 18x^{7} − 21x^{6} + 18x^{5} + 15x^{4} − 7x^{3} − 6x^{2} + x + 1 is the minimal polynomial of tr(αζ). The field I Q[θ] has a non-abelian Galois group of order 1536 = 2^{9} · 3, discriminant 7^{8} · 13^{4} · 41, classnumber 1, 4 real and 4 pairs of complex embeddings into I C and maximal order ZZ[θ]. With respect to θ and i one obtains: ξ_{7} + ξ_{7}^{−1} = 4θ^{11} − 2θ^{10} − 24θ^{9} + 15θ^{8} + 62θ^{7} − 35θ^{6} − 88θ^{5} + 14θ^{4} + 48θ^{3} + 8θ^{2} − 9θ − 4, x_{1} = \frac{1}{2}((1 + 9i)θ^{11} − (1 + 5i)θ^{10} − (6 + 54i)θ^{9} + (7 + 37i)θ^{8} + (15 + 139i)θ^{7} − (18 + 88i)θ^{6} −(21+197i)θ^{5} +(18+46i)θ^{4} +(15+111i)θ^{3} +(−7+9i)θ^{2} −(5+23i)θ+1−7i), x_{2} = \frac{1}{2}(−(5 + 3i)θ^{11} + (3 + i)θ^{10} + (30 + 18i)θ^{9} − (22 + 8i)θ^{8} − (77 + 47i)θ^{7} + (53 + 17i)θ^{6} +(109+67i)θ^{5} +(−32+4i)θ^{4} −(63+33i)θ^{3} −(1+15i)θ^{2} +(13+3i)θ +3+5i). Since tr(αζ) generates I Q[θ], the matrix αζ has infinite order. By conjugating the

representation with the matrix ( \begin{matrix}1+i & -1+i+θ^{7}-iθ^{7} \\ 0 & 2 \end{matrix} ), one obtains a representation

over ZZ[i, θ]. The enveloping algebra of this representation, i.e. the I Q-algebra generated by α, β, ζ has dimension 4 over its centre I Q[θ], therefore it is a skewfield over I Q[θ] which is the character field of the representation. ¿From the above representation one obtains homomorphisms into P SL_{2}(p) e.g. for p = 13, 197, 281, 293, 337, 349, 433, . . ..

3 The use of finite images

In this section we demonstrate how finite images H of G can be used as a guideline to construct a representation of G. Note, even for the direct methods described in Section 2 one has to choose a degree for the representation. In the insoluble case there are essentially two methods available to find finite images. The first one is described in [HoP 89], Section 7.1, cf. also [HoR 93] for a description of an implementation and further details. This method finds homomorphisms from a finitely presented group into a given permutation group and works very well for images of orders up to 10^{6}. The second method is the resultant method described in Section 2, which is rather powerful especially to obtain images P SL_{2}(q), since it more or less computes q rather than tests for solutions.

To get an idea what sort of information is useful, assume Δ : G → SL_{n}(F) is a homomorphism, where Δ(G) is infinite and F is a global field, i.e. a finite extension of I Q or of IF_{p}(x). Since G is finitely generated, the image Δ(G) must lie in an S-arithmetic subgroup of SL_{n}(R), where R is the localization of the integers in F by a multiplicative set S generated by finitely many elements. In particular, taking one of the infinitely many prime ideals ℘ of R one gets via the composition Δ_{℘} of G → SL_{n}(R) and SL_{n}(R) → SL_{n}(R/℘) a finite epimorphic image of G. Not all of these images can be trivial. If char(F) = 0, Tschebotaröw’s density theorem (cf. [Tsc 26]) tells us that ℘ can be chosen such that R/℘ ≅ IF_{p} for some prime number p. In any case let p = char(R/℘). Then the kernel K of Δ_{℘} maps under Δ into the congruence subgroup {γ ∈ SL_{n}(R)|γ − I_{n} ≡ 0 mod ℘} of SL_{n}(R) and therefore one expects X := Δ(K) and hence also K to have a substantial pro-p-completion \overline{X} := lim ← X/λ^{i}(X) resp. \overline{K} := lim ← K/λ^{i}(K), where (λ^{i}(Γ)) denotes the lower pcentral series of Γ defined by λ^{1}(Γ) := Γ and λ^{i+1}(Γ) := [λ^{i}(Γ), Γ]λ^{i}(Γ)^{p}. Instead of taking the universal pro-p-completion \overline{X} of X one can also look at the ℘-adic completion X_{℘} of X ⊆ SL_{n}(R) or even of Δ(G) itself in SL_{n}(R_{℘}), where R_{℘} := lim ← R/℘^{i}. This will also be a compact pro-p-group, namely a (continuous) image of \overline{X}. No matter whether or not X_{℘}, \overline{X} and \overline{K} are isomorphic it is reasonable to assume that X_{℘} is of finite index in a maximal compact subgroup of the group of F_{℘}-rational points of some reductive group defined over the field of fractions F_{℘} of R_{℘}. Note, the reductive groups over local fields are classified in characteristic 0 in [Kne 65] and [Sat 71], for characteristic 0 and p > 0 in [BrT 72] and [Tit 79], where also their maximal compact subgroups are investigated. For guessing a good candidate for the reductive group and the field K_{℘} it is very helpful not only to have the finite image H (assumed to be isomorphic to Δ_{℘}(G) or at least to Δ_{℘}(G)/O_{p}(Δ_{℘}(G)), but also a power commutator presentation of K/λ^{i}(K) for some i (particularly if X_{℘}, \overline{X} and \overline{K} are isomorphic). For instance if the sequence |λ^{i}(K)/λ^{i+1}(K)| = p^{c_{i}} is bounded and becomes periodic one can guess the dimension and maybe the type of the Lie algebra of the algebraic group, or if \overline{K} is powerful, the characteristic of F must be zero. But much more information such as ramification index etc. can be extracted. Having guessed the algebraic group one knows which is the smallest matrix degree to try. We note that our assumptions open up new possibilities where the existing methods - e.g. looking for a free abelian section or applying refinements of the Golod-S̆afarevic̆- bound (cf. e.g. [GaN 70]) - do not work. The next example shows that this information alone might sometimes suffice to get the direct method started, though the finite factor group SL_{2}(7) one has at the beginning turns out to be a proper epimorphic image of Δ_{℘}(G).

Example 3: The group G := ⟨a, b, c|a^{3}, b^{3}, c^{3}, ababa^{−1}b^{−1}, bcbcb^{−1}c^{−1}, cacac^{−1}a^{−1}⟩ was investigated in connection with buildings of type Ã_{2} in [Wes 85]. D. Holt checked that this group has a map onto SL_{2}(7) for which the kernel K has a lower 7-central series with a repeating sequence of layers 7^{2}, 7^{1}, 7^{2}, 7^{3} at least down to class 16 and where λ^{4}(K) is powerful. Since 2 + 1 + 2 + 3 = dim(A_{2}), this indicates that the group might have a Lie group structure of type A_{2}, hence we try a representation of degree 3 in characteristic 0. Clearly ⟨a, b⟩ ≅ ⟨b, c⟩ ≅ ⟨c, a⟩ ≅ F rob(21). Thus one may map a on the matrix

α := ( \begin{matrix}θ & -1 & -1 \\ 0 & 0 & 1 \\ -1-θ & 1 & -θ \end{matrix} ) and b on β := ( \begin{matrix}1 & 0 & 0 \\ 0 & 0 & 1 \\ θ & -1 & -1 \end{matrix} ), where θ^{2} + θ + 2 = 0, then

⟨α, β⟩ ≅ F rob(21). Let ζ := ( \begin{matrix}x_{1} & x_{2} & x_{3} \\ x_{4} & x_{5} & x_{6} \\ x_{7} & x_{8} & x_{9} \end{matrix} ) be the image of c. Since c has order

3 we must have tr(ζ) = 0. In ⟨a, c⟩ the element a^{2}c has order 3, hence tr(α^{2}ζ) = 0 and the same argument for ⟨b, c⟩ gives tr(β^{2}ζ) = 0. The elements bc and ca have order 7, hence tr(βζ) and tr(ζα) are θ or −1 − θ. Here one has to make a choice and we assume both traces to be θ. By each of these 5 linear equations one can eliminate one indeterminate and one finally comes up with a system of 6 non-linear equations in the indeterminates x_{2}, x_{3}, x_{4}, x_{6}.

Using the resultant method one obtains the solution ζ = ( \begin{matrix}-1-\frac{1}{2}θ & 1-\frac{1}{2}θ & -θ \\ \frac{1}{2}-θ & 1+\frac{1}{2}θ & 1 \\ -\frac{1}{2} & \frac{1}{2}θ & 0 \end{matrix} ).

The denominators cannot be eliminated, since tr(αβζ) = \frac{1}{2}(1 − θ), which is not an algebraic integer in I Q[θ]. This proves as well that the image is infinite. With respect to complex conjugation the above representation fixes the positive def-

inite hermitian form F := ( \begin{matrix}3 & \overline{θ} & \overline{θ} \\ θ & 3 & θ \\ θ & \overline{θ} & 3 \end{matrix} ) of determinant 7.

¿From this representation one obtains epimorphisms onto SL_{3}(p) for p ≡ 1, 2, 4 mod 7 and onto SU_{3}(p) for p ≡ 3, 5, 6 mod 7. For p = 7 one obtains the affine group Aff_{2}(7) ≅ 7^{2}><SL_{2}(7). This explains the strange pattern of the lower 7-central series of the kernel of G → SL_{2}(7), which after all is the sort of pattern one would expect for a group coming close to the pro-p-Sylow subgroup of a parahoric subgroup of a unitary group (cf. [BrT 72], [Tit 79]).

It can happen that the image H of G is too large to obtain a presentation of the kernel (e.g. if the resultant method yields an image P SL_{2}(p) for a big prime number p). In this situation one can still try to obtain some more modest information from H. The following lemma shows how to decide whether for an arbitrary epimorphic image H of G an epimorphism can be lifted to a split extension of a IF_{p}H-module by H.

Lemma 3.1 Let φ be an epimorphism of G onto H and V a simple IF_{p}H-module. Then the maximum number a such that φ can be lifted to an epimorphism onto the split extension V^{a}: H is a = dim(Der(G, V)) − dim(Der(H, V)).

Proof: This follows from [Ple 87], Section 2.

Remark 3.2 Let φ be an epimorphism of G onto H. If defining relations for H in terms of the φ(g_{i}) are known the problem of calculating the dimensions of the spaces of derivations can be reduced to a linear problem over IF_{p} by using Fox-derivatives (cf. [HoP 89], Section 4.2).

For the rest of this section we assume that G has a homomorphism φ onto P SL_{2}(p) for some prime p. That this is not too much a restriction follows from the following two observations:

1) “Most” simple groups are of this type (e.g. 50% of the groups of order less than 10^{6}).

2) If G has a 2-dimensional representation over any field of characteristic 0 then it follows from Tschebotaröw’s density theorem (cf. [Tsc 26]), that there are infinitely many primes p such that G maps into P SL_{2}(p).

The next lemma gives a lower bound for the degree of the representation to construct in case the field is of characteristic p or ramified over I Q_{p}.

Lemma 3.3 Assume that V is a simple IF_{p}P SL_{2}(p) module of dimension d > 1 such that φ can be lifted to the split extension V : P SL_{2}(p). Then this lift can not be realized in P SL_{k}(IF_{p}[x]/x^{2}IF_{p}[x]) for k < (d + 1)/2.

Proof: Denote by V_{k} the k-dimensional simple IF_{p}SL_{2}(p) module and by Δ_{k} the corresponding representation of SL_{2}(p). Then the module for the action of Δ_{k}(SL_{2}(p)) on (xIF_{p}[x]/x^{2}IF_{p}[x])^{k×k} is the tensor square of V_{k}, since V_{k} is selfdual. But the largest simple composition factor of V_{k}⊗V_{k} is V_{2k−1} as an easy calculation with Brauer characters shows.

We now want to decide whether an epimorphism lifts to a non-split extension V.P SL_{2}(p) for a simple IF_{p}P SL_{2}(p)-module V.

Lemma 3.4 Assume p > 3 and let V_{i} be the simple IF_{p}P SL_{2}(p)-module of dimension i. Then

i) H^{2}(P SL_{2}(p), V_{i}) = {0} if i ≠ 3.

ii) H^{2}(P SL_{2}(p), V_{3}) ≅ IF_{p}.

Proof: Let V be any finite IF_{p}P SL_{2}(p)-module, I(V) its injective hull and Ω^{−1}V the cokernel of V → I(V), i.e. the inverse of the Heller operator. Since I(V) is projective, Ĥ^{i}(I(V)) = 0, and therefore by the long exact sequence for Tate cohomology Ĥ^{i}(V) ≅ Ĥ^{i−1}(Ω^{−1}V) (cf. [CuR 90] §8D). In particular H^{2}(V) ≅ Ĥ^{2}(V) ≅ Ĥ^{0}(Ω^{−2}V). One can easily read off Ω^{−2}V_{i} from the Brauer tree (cf. [Alp 86]):

[figure: V_{1}; V_{p−2}; V_{3}; V_{p−4}; V_{(p+ε)/2}]

where ε = ±1 with p ≡ ε mod 4 and the multiplicity of the exceptional vertex is 2. Clearly H^{2}(V_{1}) = {0}, since the Schur multiplier of P SL_{2}(p) is isomorphic to C_{2}, H^{2}(V_{p}) = {0}, since V_{p} is injective and Ĥ^{0}(Ω^{−2}V_{n}) = {0} if Ω^{−2}V_{n} does not have V_{1} in its socle. The only critical cases are the modules V_{p−2} and V_{3}. In

obvious notation giving the socle series one has: Ω^{−2}V_{p−2} = \begin{matrix}V_{1} V_{3} \\ V_{p−2} V_{p−4} \end{matrix} and

Ω^{−2}V_{3} = \begin{matrix}V_{p−2} V_{p−4} \\ V_{1} V_{3} V_{5} \end{matrix} for p > 5 resp. Ω^{−2}V_{3} = \begin{matrix}V_{1} V_{3} \\ V_{3} V_{1} \end{matrix} for p = 5. In

particular Ĥ^{0}(Ω^{−2}V_{i}) = {0} for i ≠ 3 and Ĥ^{0}Ω^{−2}V_{3} ≅ IF_{p}, since in our situation Ĥ^{0}(Ω^{−2}V_{3}) ≅ H^{0}(Ω^{−2}V_{3}).

Lemma 3.5 For p > 3 an epimorphism φ of G onto P SL_{2}(p) lifts to a non-split extension V_{3}.P SL_{2}(p) if and only if the projective representation of φ of G can be lifted to a projective representation of G onto P SL_{2}(ZZ_{p}/p^{2}ZZ_{p}).

Proof: This follows immediately from the preceeding lemma and Theorem 2.3.37 in [HoP 89] (cf. also [Wal 68]).

4 The lifting procedure

For this section we fix the following notation: let R be a complete discrete valuation ring with maximal ideal ℘ = πR, residue class field IF_{q} and field of fractions K, which is a finite extension of I Q_{p} or IF_{q}((x)) with ramification index e in the first case. Set e = ∞ in the second case. We assume that we have a homomorphism φ from G into P SL_{k}(IF_{q}). We want to lift φ to a homomorphism of G into P SL_{k}(R) such that φ is the composition of that homomorphism with the natural projection from R to R/℘.

First we describe how φ can be lifted (if such a lift is possible) to a homomorphism into P SL_{k}(R/℘^{c}) for some finite c > 1 by solving linear equations over IF_{q}.

Lemma 4.1 Let f_{1}, …, f_{m} ∈ R[X_{1}, …, X_{n}] and assume that for y = (y_{1}, …, y_{n}) ∈ R^{n} one has f_{i}(y) ≡ 0 mod ℘^{d} for 1 ≤ i ≤ m. Let z = (z_{1}, …, z_{n}) ∈ R^{n} and define e′ := min(d, e). Then the conditions f_{i}(z) ≡ 0 mod ℘^{d+e′} for 1 ≤ i ≤ m and z_{j} ≡ y_{j} mod ℘^{d} for 1 ≤ j ≤ n are equivalent to a certain IF_{q}-linear system of equations in π^{−d}(z_{j} − y_{j}) mod ℘^{e′}.

Proof: For 1 ≤ j ≤ n write z_{j} = y_{j} + π^{d}r_{j} with r_{j} ∈ R. Expanding f_{i}(z) around y and omitting terms divisible by π^{2d} yields f_{i}(y) + π^{d} ∑_{j=1}^{n} l_{ij}r_{j} ≡ 0 mod ℘^{2d}, where l_{ij} ∈ R. These are linear congruences for the r_{j}. Dividing by π^{d} and reducing modulo ℘^{e′} gives ∑_{j=1}^{n} l_{ij}r_{j} ≡ −π^{−d}f_{i}(y), which can be interpreted as IF_{q}-linear equations, since R/℘^{e′} is an IF_{q}-algebra.

Remark 4.2 (i) Obviously one has quadratic convergence for the characteristic p > 0 case. The same can be achieved in characteristic 0, if one is prepared to work in residue class rings of increasing size. (ii) It pays, however, to lift by more than one step if possible, because not every lift from R/℘^{d} to R/℘^{d+1} can be lifted further to R, but it might be that for h > 1 for every lift from R/℘^{d} to R/℘^{d+h} the restriction to R/℘^{d+1} can be lifted to R. This trick - an analogue of the famous “Echternacher Springprozession” - was already used by M. Wursthorn (cf. [Wur 93]) to calculate automorphisms of p-groups.

Concerning (non-)uniqueness of the lifting one should be aware of the following. ¿From one homomorphism φ of G into P SL_{k}(R/℘^{d+1}) one obtains further ones by conjugating the image with matrices ξ from SL_{k}(R). If ξ ≡ I_{k} mod ℘^{d} the homomorphisms φ and ξφξ^{−1} agree modulo ℘^{d} and can be viewed as lifting of the same homomorphism into P SL_{k}(R/℘^{d}). We give more precise details for the lifting from R/℘^{d} to R/℘^{d+1}. Besides the homomorphism φ_{d} : G → P SL_{k}(R/℘^{d}) one has γ_{1}, …, γ_{s} as images of g_{1}, …, g_{s} in SL_{k}(R) in the notation of Section 2. Let L(φ_{d}) be the set of all homomorphisms φ : G → P SL_{k}(R/℘^{d+1}) with g_{i}φ ≡ γ_{i} mod ℘^{d+1} for i = 1, …, s and g_{i}φ ≡ g_{i}φ_{d} mod ℘^{d} for i = s + 1, …, n (ignoring the scalar matrices to be factored out). L(φ_{d}) can be viewed as an affine space over R/℘ = IF_{q}, because of the previous lemma. Let C := {X ∈ R^{k×k}|Xγ_{i} ≡ γ_{i}X mod ℘ for i = 1, …, s} and C_{0} := {X ∈ C|Xg_{i}φ_{d} ≡ g_{i}φ_{d}X mod ℘ for i = s + 1, …, n}. Note, both C and C_{0} only depend on φ_{d} modulo ℘ (and are therefore constant throughout the computation). C acts on L(φ_{d}) as follows: c ∈ C maps φ ∈ L(φ_{d}) to φ^{[c]}, which is the composition of φ with conjugation by I_{k} + π^{d}c. The crucial observation is

g_{i}φ^{[c]} ≡ g_{i}φ + π^{d}(c(g_{i}φ) − (g_{i}φ)c) mod ℘^{d+1} (and modulo scalar matrices).

One easily verifies:

Lemma 4.3 Let C act on L(φ_{d}) as described above.

i) C_{0} is the kernel of this action, more precisely each C-orbit on L(φ_{d}) is in bijection with the IF_{q}-vector space C/C_{0}.

ii) The C-action respects liftability (to R/℘^{d+2} or even to R).

Thus let α := dimIF_{q}C/C_{0} and let L^{h}(φ_{d}) be the set of all φ ∈ L(φ_{d}) which lift to a homomorphism into P SL_{k}(R/℘^{d+h}). L^{h}(φ_{d}) is an affine subspace of L(φ_{d}) = L^{1}(φ_{d}) for h ≤ d. Denote its dimension by β_{h}. If α < β_{1} there is the danger that not everything lifts, i.e. L^{h}(φ_{d}) ≠ L(φ_{d}) for some h > 1. In this situation Remark 4.2(ii) should be taken serious.

The following lemma is a multi-dimensional analogue of Hensel’s lemma. It gives an explicit value d_{0} such that an epimorphism into P SL_{k}(R/℘^{d}) can be lifted to P SL_{k}(R) for each d > d_{0}.

Lemma 4.4 ([Bou 72], Chapter III, §4.6, Corollary 3) Let f_{1}, …, f_{m} ∈ R[X_{1}, …, X_{n}] where n ≥ m. Denote by M_{f} the Jacobian matrix (∂f_{i}/∂X_{j}) and by J_{f}^{(m)}(X_{1}, …, X_{n}) the determinant of the last m columns of M_{f}. Let y = (y_{1}, …, y_{n}) ∈ R^{n} and assume that f_{i}(y) ≡ 0 mod ℘^{1+2d} for 1 ≤ i ≤ m, where d is such that ℘^{d} = J_{f}^{(m)}(y)R. Then there exists z = (z_{1}, …, z_{n}) ∈ R^{n} such that z_{j} ≡ y_{j} mod ℘^{1+2d} for 1 ≤ j ≤ n and f_{i}(z) = 0 for 1 ≤ i ≤ m.

This lemma can be applied to a system of equations where the number of equations is not bigger than the number of indeterminates. But the reduction of the number of equations as outlined in Section 2 often suffices to fulfil this hypothesis. However, in all examples we have treated up to now, we could do in the end without this lemma, because we succeeded to pass from the representation modulo ℘^{c} to a global representation as discussed in the next section.

5 From local to global representations

Since the entries of a representation of G are solutions of a system of algebraic equations, one can usually arrange for a representation in characteristic 0 that it can be realized over some algebraic number field K. The required extension of I Q can be determined from the p-adic expansion of the solutions by the following method (cf. [LLL 82], [dWe 89]):

Lemma 5.1 Let x = ∑_{i=0}^{∞} a_{i}p^{i} be a p-adic number which is algebraic over I Q and denote the n-th partial sum of x by x_{n}. Define L_{m,n} := {v = (v_{0}, v_{1}, …, v_{m}) ∈ ZZ^{m+1}| ∑_{i=0}^{m} v_{i}x_{n}^{i} ≡ 0 mod p^{n+1}} and let L_{m} := ∩_{n=0}^{∞}L_{m,n}. Then [ I Q[x] : I Q] = min{m ∈ IN | L_{m} ≠ {0}}.

Proof: Let m_{x} = ∑_{i=0}^{d} c_{i}t^{i} be the minimal polynomial for x. Then clearly the vector (c_{0}, …, c_{d}, 0, …, 0) lies in L_{m} for all m ≥ d. Suppose now m < d and v ∈ L_{m}. Then for all n ∈ IN one has ∑_{i=0}^{m} v_{i}x_{n}^{i} ≡ 0 mod p^{n+1}, hence ∑_{i=0}^{m} v_{i}x^{i} = 0, which implies v = 0 since m < d.

The vector v_{x} of coefficients of the minimal polynomial of x can be calculated using the LLL-algorithm (cf. [LLL 82]) as follows: The lattice L_{m,n} has index p^{n+1} in ZZ^{m+1} and a ZZ-basis of L_{m,n} is (e_{0}, e_{1}, …, e_{m}), where e_{0} = (p^{n+1}, 0, …, 0) and e_{i} = (−x_{n}^{i}, …, 1, …) with the 1 at position i. If m = d the vector v_{x} lies in L_{m,n} for all n. Now for growing n the Euclidean norms of the vectors in L_{m,n} which are linearly independent from v_{x} increase. Thus for n sufficiently large v_{x} can be found as a vector which is much shorter than all others by means of the LLL-algorithm. In our examples a faster pair-reduction applying usual Minkowski-reduction for dimension 2 to every pair of basis vectors yields already the desired shortest vector. The above method can be used as well to express an algebraic number y in terms of powers of x. For that one replaces the basis vector e_{m} by (−y, …, 1). Then the LLL-algorithm yields the coefficients of a linear combination of 1, x, …, x^{m−1} representing y.

Example 4: Let G be given by ⟨a, b|a^{3}, b^{3}, baba^{−1}ba^{−1}b^{−1}abab^{−1}ab^{−1}a^{−1}⟩. This group was on the one hand investigated by G. Rosenberger and L. Lévai in connection with generalized triangel groups and on the other hand by V. Metaftsis as a group with two generators of order 3 and one additional short relator. Its infiniteness was still open. D. Holt checked that G has a map onto SL_{2}(5) for which the kernel has a lower 5-central series with a repeating sequence of layers 5^{2}, 5^{1}, 5^{2}, 5^{3} at least down to class 16. This indicates that the group is connected with a Lie group of type A_{2}. Furthermore the structure of the central series implies that some subgroup is powerful. Hence we try a representation of degree 3 in characteristic 0. In this case the system of algebraic equations is too complicated for the direct methods. After fixing the image of a and using the trace of b there remain 8 indeterminates. The remaining 2 coefficients in the characteristic polynomial of b give 2 nice equations. If one uses the characteristic polynomial of the third relator one obtains 2 further equations with 222 resp. 5800 terms and degree 5 resp. 10. Alternatively one can use the entries of the third relator, then one gets 9 further equations with about 100 terms and degree 5. Instead we look for primes p such that G maps onto SL_{3}(p). Of course G maps into SL_{3}(5) with various homomorphisms factoring over SL_{2}(5). Indeed the situation in the end turns out to be analogous to Example 3 at p = 7. It is more convenient to

work with the prime 11, namely G maps onto SL_{3}(11) by a ↦ α := ( \begin{matrix}0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{matrix} )

and b ↦ β_{0} := ( \begin{matrix}0 & 0 & 1 \\ -2 & 1 & 5 \\ -1 & 0 & -1 \end{matrix} ). This modular solution can be lifted to a solution β

over the ring ZZ_{11} of 11-adic integers. In each step of the lift one has 3 free parameters. Choosing the first row of β as (0, 0, 1) the lift becomes unique. By the method described in Lemma 5.1 one now calculates that θ := tr(αβα) has minimal polynomial m_{θ} = x^{6} − 3x^{3} + 1 and that the other entries of β can be expressed in terms of θ. The field I Q[θ] has Galois group D_{12}, classnumber 1, discriminant 3^{6} · 5^{3}, 2 real and 2 pairs of complex embeddings into I C and maximal order ZZ[θ]. The fact that (αβ)^{2} ≠ 1 and tr((αβ)^{2}) = 3 proves that the image is infinite. To write the representation over ZZ[θ] one has to conjugate with a suitable matrix and finally obtains:

a ↦ ( \begin{matrix}-3θ^{4}+8θ & -4θ^{4}+11θ & 2θ^{3}-6 \\ 0 & 0 & 1 \\ -5θ^{5}+14θ^{2} & -7θ^{5}+19θ^{2} & 3θ^{4}-8θ \end{matrix} ), b ↦ ( \begin{matrix}-5θ^{5}+14θ^{2} & -7θ^{5}+19θ^{2} & 3θ^{4}-8θ \\ 3θ^{5}-9θ^{2} & 5θ^{5}-14θ^{2} & -2θ^{4}+6θ \\ 1 & 0 & 0 \end{matrix} ).

Denote by the Galois automorphism of I Q[θ] that maps θ on θ^{−1} = −θ^{5} + 3θ^{2}. With respect to this automorphism the above representation fixes the hermitian

form ( \begin{matrix}4 & θ^{3}-5 & \overline{θ} \\ \overline{θ^{3}}-5 & 4 & \overline{θ} \\ θ & θ & 4 \end{matrix} ) of determinant 5.

¿From this representation one obtains epimorphisms onto SL_{3}(p) for p = 11, 29, 41, 59, 71, 89, … and onto SU_{3}(p) for p = 3, 7, 13, 37, 43, 47, 67, 73, 97, … depending on the fact whether the polynomial m_{θ} has irreducible linear or quadratic factors over IF_{p}. For p = 5 one obtains the affine group Aff_{2}(5) ≅ 5^{2}><SL_{2}(5).

In characteristic p one will try to construct from a representation over IF_{p}[[x]] one over the field of rational functions over IF_{p}. The following lemma, which was probably already known to Frobenius (cf. e.g. [Gan 59], Chapter XV, §10), gives an easy method for this.

Lemma 5.2 There exists a bijection between periodic power series expansions in 0 over IF_{p} and rational functions over IF_{p} not having a pole in 0.

Proof: Let r := \frac{g}{f} be a rational function in x, gcd(f, g) = 1, p a prime such that p| f̸(0). If f is irreducible over IF_{p} one has f | x^{q−1} − 1, where q = deg(f). If f = h^{b} where h is irreducible and b ∈ IN it follows for q = deg(h) and p^{c} ≥ b that f | x^{(q−1)p^{c}} − 1, since (x^{q−1} − 1)^{p^{c}} = x^{(q−1)p^{c}} − 1 in characteristic p. Finally assume f_{1}, f_{2} ∈ IF_{p}[x], a, b ∈ IN such that gcd(f_{1}, f_{2}) = 1 and f_{1} | x^{a} − 1 and f_{2} | x^{b} − 1. Then f_{1}f_{2} | x^{lcm(a,b)} − 1. Hence it follows by induction that r can be written as \frac{gh}{1−x^{a}} for some a ∈ IN and h ∈ IF_{p}[x] which gives a periodic expansion of r by means of the geometric series. The opposite direction follows immediately from the geometric series. The periodic expansion given by the coefficients (a_{0}, a_{1}, …, a_{s}, \overline{b_{1}, …, b_{n}}) with a_{i}, b_{j} ∈ IF_{p} is the

power series expansion of the function r = ∑_{i=0}^{s} a_{i}x^{i} + x^{s+1} \frac{∑_{j=1}^{n} b_{j}x^{j−1}}{1 − x^{n}} .

The following example was already studied in [Wes 85]. We want to show how a representation of this group can be obtained by our method of ℘-adic expansion.

Example 5: Let G := ⟨a, b, c|a^{3}, b^{3}, c^{3}, aba^{−1}ba^{−1}b^{−1}, bcb^{−1}cb^{−1}c^{−1}, aca^{−1}ca^{−1}c^{−1}⟩. The commutator subgroup G′ has index 3 in G and G′′ has index 7^{2} in G′. Applying the nilpotent quotient algorithm for p = 2 to G′′ gives a recurring sequence of layers 2^{6}, 2^{6}, 2^{4}. This indicates a Lie group structure of type A_{2} ×A_{2}. Furthermore a closer analysis of the central series shows that this Lie algebra must have characteristic 2. We therefore try a representation of degree 3 over the field IF_{2}[[x]] of Laurent series over IF_{2}. Clearly ⟨a, b⟩ ≅ ⟨b, c⟩ ≅ ⟨c, a⟩ ≅ F rob(21). Thus one may map a on the matrix

α := ( \begin{matrix}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 1 \end{matrix} ) and b on β := ( \begin{matrix}1 & 1 & 1 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{matrix} ), then ⟨α, β⟩ ≅ F rob(21). Let ζ :=

( \begin{matrix}x_{1} & x_{2} & x_{3} \\ x_{4} & x_{5} & x_{6} \\ x_{7} & x_{8} & x_{9} \end{matrix} ) be the image of c. Like in Example 3 one concludes that tr(ζ) =

tr(αζ) = tr(βζ) = 0 which gives the equations x_{9} = x_{1} + x_{5}, x_{8} = x_{5} + x_{6} and x_{7} =

x_{2} + x_{4} + x_{5}. A solution modulo the maximal ideal of IF_{2}[[x]] is ζ_{0} := ( \begin{matrix}0 & 1 & 1 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{matrix} ).

This solution can be lifted to a solution ζ in SL_{3}(IF_{2}[[x]]). Furthermore it turns out that this is possible in such a way that the entries of ζ have periodic expansions. By the method described in Lemma 5.2 one now finds the following solution over the field

of rational functions over IF_{2}: ζ := u^{−1} ( \begin{matrix}0 & u & u(1 + x) \\ 1 & x^{2} & 1 + x^{2} \\ x & 1 & x^{2} \end{matrix} ), where u := 1 + x + x^{2}.

The matrix αβζ has infinite order, since tr(αβζ) = u^{−1}(1 + x)^{3}.

6 Bibliography

[Alp 86] J.L. Alperin, Local Representation Theory. Cambridge University Press, Cambridge, 1986

[Bou 72] N. Bourbaki, Commutative Algebra. Hermann, Paris, 1972 [BrT 72] F. Bruhat, J. Tits, Groupes réductifs sur un corps local. IHES, Publications

Mathématics No. 41, pp. 5-252, 1972

[Buc 65] B. Buchberger, Ein Algorithmus zum Auffinden der Basiselemente des Restklassenrings nach einem nulldimensionalen Polynomideal. Doctoral Dissertation Math. Inst. University of Innsbruck, Austria, 1965

[CuR 90] C.W. Curtis, I. Reiner, Methods of Representation Theory, Volume I. Wiley, New York, 1990

[Gan 59] F.R. Gantmacher, Matrizenrechnung II. VEB Deutscher Verlag der Wissenschaften, Berlin, 1959

[GaN 70] W. Gaschütz, M.F. Newman, On presentations of finite p-groups. Journal für reine angew. Math., 245, pp. 172-176, 1970

[GCL 92] K.O. Geddes, S.R. Czapor, G. Labahn, Algorithms for Computer Algebra. Kluwer Academic Publishers, Boston, 1992

[HaN 80] G. Havas, M.F. Newman, Application of computers to questions like those of Burnside. Lecture Notes in Mathematics 806, pp. 211-230, Springer, Berlin, 1980

[HoP 89] D.F. Holt, W. Plesken, Perfect Groups. Oxford University Press, Oxford, 1989

[HoP 92] D.F. Holt, W. Plesken, A Cohomological Criterion For A Finitely Presented Group To Be Infinite. Journal London Math. Soc. (2), 45, pp. 469-480, 1992

[HoP 93] D.F. Holt, W. Plesken, The p-Coclass of a Group. Journal of Algebra, Vol. 162, No. 2, pp. 335-342, 1993

[HoR 93] D.F. Holt, S. Rees, A graphics system for displaying finite quotients of finitely presented groups. Groups and Computation, DIMACS Series in Discrete Mathematics and Theoretical Computer Science 11, AMS, pp. 113-126, 1993

[HPS 94] D.F. Holt, W. Plesken, B. Souvignier, Constructing a representation of the group (2, 3, 7; 11). These proceedings

[Kne 65] M. Kneser, Galois-Kohomologie halbeinfacher algebraischer Gruppen über padischen Körpern. Math. Z. 88, pp. 40-47, 89, pp. 250-272, 1965

[LLL 82] A.K. Lenstra, H.W. Lenstra, L. Lovász, Factoring polynomials with rational coefficients. Math. Annalen 261, pp. 515-534, 1982

[Ple 87] W. Plesken, Towards a Soluble Quotient Algorithm. Journal Symbolic Computation 4, pp. 111-122, 1987

[Poh 93] M. Pohst e.a, KANT-V2. Computer Algebra in Deutschland, edited by Fachgruppe Computeralgebra der GI, DMV und GAMM, pp. 212-218, 1993 [Sat 71] I. Satake, Classification Theory of Semi-Simple Algebraic Groups. Lecture

Notes in Pure and Applied Mathematics, New York, 1971

[Tit 79] J. Tits, Reductive groups over local fields. AMS Proceedings of Symposia in Pure Mathematics, Vol. 33, part 1, pp. 29-69, 1979

[Tsc 26] N. Tschebotaröw, Die Bestimmung der Dichtigkeit einer Menge von Primzahlen, welche zu einer gegebenen Substitutionsklasse gehören. Math. Annalen 95, pp. 191-228, 1926

[Wal 68] G.E. Wall, A characterization of P SL_{2}(ZZ_{p^{x}}) and P GL_{2}(ZZ_{p^{x}}). Journal Australian Math. Soc. 8, pp. 523-543, 1968

[dWe 89] B. de Weger, Algorithms for diophantine equations. CWI TRACT 65, pp. 60-69, 1989

[Wes 85] M. Wester, Endliche fahnentransitive Tits-Geometrien und ihre universellen Überlagerungen. Mitteilungen aus dem Math. Seminar Giessen, Heft 170, pp. 62-73, 1985

[Wur 93] M. Wursthorn, Isomorphisms of modular group algebras: An algorithm and its application to groups of order 2^{6}. Journal Symbolic Computation 15, pp. 211-227, 1993

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