Algorithms determining finite simple images of finitely presented groups

3.1 Actions on adjoint modules

The main result we shall need is Theorem 3.1, most of which follows from [17, Theorem 4] and [18].

Recall that \({\bar{X}}\) acts on its adjoint module, the Lie algebra \(L({\bar{X}})\). This action is usually irreducible, but there are exceptions in a small number of characteristics; the precise composition factors can be found in [17, 1.10].

We also require the following fact about containment between the groups \(^d\!X(q)\).

For \(d= 1,2,3\) we say that the d-part of \(e \in {\mathbb {N}}\) is the largest power of d which divides e.

3.2 Model theory

In this section we present some results from model theory that we need. We work with the usual language of rings \(L = \{0, 1, +, -, \cdot \}\). ‘Definable’ means ‘definable without parameters’ unless explicitly stated otherwise.

In [1], Ax showed that there is an algorithm to decide, given a closed L-formula, whether or not there are infinitely many finite fields in which it holds. The algorithm makes use of Gödel’s completeness theorem and involves an unbounded (but terminating) search. A primitive recursive algorithm (so not involving unbounded searches) was given by Fried and Sacerdote in [8]. (Recursive procedures for the other results below had also been given by Ax.) We state their results in the following way:

Note that (i) says that we can determine whether or not \(\{q \in {\mathcal {Q}}: {\mathbb {F}}_q \models \lnot \theta \}\) is small. As a special case of (i), we can determine whether or not \(\theta \) is true in all finite fields.

We also require the following result, which follows from Theorem 6.7 of [6].

The following is the effective version of the main result of [5]. Here, ‘effective’ means computable in a primitive recursive way.

In the above, we may assume that the formula \((\forall \bar{x})(\lnot \theta (\bar{x},\bar{y}))\) is one of the \(\theta _i(\bar{y})\). This then picks out those \(q, \bar{b}\) where \(N_q(\bar{b})\) is zero.

3.3 Deduction of Theorem 2

For a field K, let \(\mathrm{ID}(X(K))\) denote the inner-diagonal group of (untwisted) type X over K, and let \(V = V(K)\) be the adjoint module for \(\mathrm{ID}(X(K))\). For \(K = {\mathbb {F}}_q\), we can regard \(\mathrm{ID}(X(K))\) as a definable subset H(K) of \(K^{l^2}\) (where \(l = \dim V\)) in a uniform way via the action on V, as the solutions of polynomial equations defining the connected adjoint algebraic group of type X.

If \(X = A_n\), \(D_n\) or \(E_6\) and \(K = {\mathbb {F}}_q\), then X possesses a graph automorphism of order \(d \in \{2,3\}\), and the twisted groups \(\mathrm{ID}(^d\!X(q))\) can also be regarded as uniformly definable subsets \(H({\mathbb {F}}_q)\) of some power of K, using Remark 3.8. This is covered in detail in Section 5.3 of [24].

In one exceptional case we vary the above definition of H(K) and V(K). This is the case where \(X = C_n\,(n\ge 2)\) and \(\mathrm{char}(K) = 2\). In this case we have \(\mathrm{ID}(X(K)) = X(K)\); let \(V_{2n}(K)\) be the natural module, and S(K) a spin module for X(K), and for \(K = {\mathbb {F}}_q\) regard X(K) as a uniformly definable subset H(K) of \(K^{l^2}\) via the action on \(V = V_{2n}(K)\oplus S(K)\), where \(l = 2n+2^n\).

For a tuple \({\bar{g}} = (g_1,\ldots , g_s) \in \Phi _0(K)\), define \(\langle {\bar{g}}\rangle : = \langle g_1,\ldots ,g_s\rangle \). There is an L-formula that expresses the condition that the subgroup \(\langle {\bar{g}}\rangle \) of \(\mathrm{GL}(V)\) is absolutely irreducible on every H(K)-composition factor of V. One way of writing down such a formula is as follows. Fix a basis B of V. Let \(K_1\) be an extension field of K with \(|K_1:K|\le n\) (definable in K as in Remark 3.8), let \(V(K_1)\) be the \(K_1\)-vector space \(V \otimes _K K_1\), and let \({{\mathcal {W}}}\) be the collection of subspaces of \(V(K_1)\) that are spanned by some proper non-empty subset of B. The formula says that for any such \(K_1\), there is no pair \(h \in \mathrm{GL}(V(K_1))\), \(W \in {{\mathcal {W}}}\) such that each of the conjugates \(g_i^h\) stabilizes W. There is also a formula expressing the condition \(|\langle {\bar{g}}\rangle |>f(r)\), where f is the function in Theorem 3.1. Hence if we define \(\Phi (K)\) to be the set of all tuples \({\bar{g}}\in \Phi _0(K)\) such that \(\langle {\bar{g}}\rangle \) is absolutely irreducible on every H(K)-composition factor of V and has order greater than f(r), then \(\Phi (K)\) is uniformly definable in K. We can regard \(\Phi \) here as a parameter-free L-formula (with the parameters needed to define the required field extensions absorbed into the formula by making use of Remark 3.8). In principle, we could write down this formula \(\Phi \) explicitly.

In the case where \(X = C_n\) and \(\mathrm{char}(K) = 2\), we adjust \(\Phi \) to include the condition that subgroups of index at most 4 in \(\langle {\bar{g}}\rangle \) arising from subgroups of such index in G are also absolutely irreducible on every composition factor of V.

By Theorem 3.1 and Lemma 3.2, for any finite field \(K = {\mathbb {F}}_q\), and any tuple \({\bar{g}} \in \Phi (K)\), the group \(\langle {\bar{g}}\rangle \) is of the same type as X, hence is a group in the class \({{\mathcal {Y}}}\) over some subfield \({\mathbb {F}}_{q_0}\) of \({\mathbb {F}}_q\). Summarising:

At this point we apply Theorem 3.7 (taking \(m=0\)) to the parameter-free formula \(\Phi (\bar{x})\) and obtain (effectively) a finite set \(\{(\theta _i, \mu _i, \varepsilon _i, r_i): i \in I\}\) as in the theorem. Note that as there are no parameter variables in \(\Phi \), the \(\theta _i\) are L-sentences. Informally, this expresses \(\Phi ({\mathbb {F}}_q)\) as a finite union of subsets of dimensions \(r_i\,(i\in I)\), for any \(q \in {\mathcal {Q}}\).

On the other hand, we have the following simple observation.

Define \({\mathcal {Q}}(G)\) to be the set of prime powers q such that G has as a homomorphic image a group in the class \({{\mathcal {Y}}}\) over \({\mathbb {F}}_q\). Let \(S = \{q : {\mathbb {F}}_q \models \exists \bar{x}\Phi (\bar{x})\}\), and for a prime p, let \(S_p = \{e : {\mathbb {F}}_{p^e} \models \exists \bar{x}\Phi (\bar{x})\}\).

At this point we can complete the proof of Theorem 2 (apart from the Suzuki and Ree families, postponed until the next subsection). We assume Theorem 4, which will be proved in Sect. 4. Fix a Lie type \(^d\!X\), and let \({{\mathcal {X}}}^d\), \({{\mathcal {Y}}}^d\) be the corresponding classes of groups defined in the Introduction. If \(d>1\), then \({{\mathcal {Y}}} = {{\mathcal {Y}}}^d= {{\mathcal {X}}}^d\), so Corollary 3.12 shows that there is an algorithm that determines whether or not G has infinitely many quotients in \({{\mathcal {X}}}^d\). Now suppose \(d=1\), so that \({{\mathcal {Y}}} = \bigcup _e {{\mathcal {Y}}}^e\). By Corollary 3.12, there is an algorithm that determines whether or not G has infinitely many quotients in \({{\mathcal {Y}}}\). If there are infinitely many such quotients, then Theorem 4 shows that there are also infinitely many quotients in \({{\mathcal {X}}}^1\).

This completes the proof of Theorem 2 apart from the last assertion—that if the set \({\mathcal {Q}}(G)\) is finite, we can compute this set. So suppose \({\mathcal {Q}}(G)\) is finite. Note that if N is a positive integer, we can write down a sentence \(\Psi _N\) which, in a field \({\mathbb {F}}_q\), says that there is a tuple in \(H({\mathbb {F}}_q)\) which satisfies the formula \(\Phi (\bar{x})\) and which generates a subgroup of \(H({\mathbb {F}}_q)\) with at least N elements. As in Lemma 3.9, we have that \(\Psi _N\) is true in \({\mathbb {F}}_q\) if and only if there is a homomorphic image of G in \({{\mathcal {Y}}}\) of size at least N. So by our assumption, there is some N with the property that \(\Psi _N\) is false in all finite fields. Using Theorem 3.5, we can find such an N by testing whether \(\Psi _n\) is false in all finite fields for successive values of n until one is found (this involves an unbounded search). This gives a bound N on the size of the groups in \({\mathcal {Y}}\) which are homomorphic images of G, so we can test groups in \({{\mathcal {Y}}}\) of size at most N individually (a series of finite computations) to determine whether or not they are homomorphic images of G.

In Sect. 3.5, we give a more complicated argument which avoids the unbounded search in the previous paragraph, hence providing a primitive recursive algorithm for determining \({\mathcal {Q}}(G)\) in the case where it is finite.

3.4 The Suzuki and Ree groups

We are very grateful to Ivan Tomašić for his help with the material of this section. The aim is to adapt the above arguments to handle the classes \({\mathcal {X}}^2\) where X is \(B_2\), \(G_2\), or \(F_4\) (and the characteristic is 2, 3 or 2 respectively). So these are the classes consisting of the Suzuki and Ree groups \(^2\!B_2(q),\,^2\!G_2(q),\,^2\!F_4(q)\)—note that for these groups, \(\mathrm{ID}(^2\!X(q)) = \,^2\!X(q)\), so the classes consist only of the simple groups.

The appropriate model-theoretic framework is that of algebraically closed fields with a Frobenius automorphism [10], rather than the model theory of finite fields used in the previous subsection.

Let \(L^\sigma \) be the first-order language consisting of L, the language of rings used previously, together with a unary function symbol \(\sigma \). If q is a power of a prime number p and K is an algebraically closed field of characteristic p, we denote by \(K_q\) the \(L^\sigma \)-structure \((K; 0,1,+,- , \cdot , \sigma )\) where \(\sigma : K \rightarrow K\) is \(x \mapsto x^q\). In particular, the fixed field of \(\sigma \) is \({\mathbb {F}}_q\), so the field \({\mathbb {F}}_q\) is uniformly definable in \(K_q\) (that is, using an \(L^\sigma \)-formula which is independent of q). Similarly, for fixed p, let \(\nu \) denote the Frobenius map \(x \mapsto x^p\). If \(q = p^n\) then the field \({\mathbb {F}}_{p^{2n+1}}\) is uniformly interpreted in \(K_q\) as the fixed field of \(\nu \circ \sigma ^2\). Moreover, \((\nu \circ \sigma )^2(x) = \nu (x)\) for all \(x \in {\mathbb {F}}_{p^{2n+1}}\). Thus we have a uniform interpretation of the fields \({\mathbb {F}}_{p^{2n+1}}\) together with a square root of the Frobenius map on these in the structures \(K_{p^n}\).

It follows that if \(^2\!X(p^{2n+1})\) is a family of Suzuki or Ree groups, then there is a (parameter-free) \(L^\sigma \)-formula H such that, for an algebraically closed field K of characteristic p (equal to 2 or 3, as appropriate), the solution set \(H(K_{p^n})\) is the subgroup \(^2\!X(p^{2n+1})\) of the group X(K) (represented by its action on its adjoint module, as in the previous section).

The algorithm to determine whether our finitely presented group G has infinitely many images in the X-class \({\mathcal {X}}^2\) is then exactly as before, except that we require results for the \(L^\sigma \)-structures \(K_q\) in place of Theorem 3.5(ii) and Theorem 3.7. The following result (Theorem 1.1 of [25]) has been communicated to us by Ivan Tomašić.

Note that in the above, the obvious conventions about \(\infty \) are being used when \(N_q(\bar{b})\) is infinite. The paper [25] makes no claims about determining C, D and the formulas \(\varphi _{d,\mu }\). However, Tomašić informs us that, using the methods of [28], a version of the above can be proved in which these are determined in a primitive recursive way (so providing a replacement for Theorem 3.7). Moreover, in the case where there are no parameter variables \(\bar{y}\), the \(\varphi _{d,\mu }\) will be such that testing whether or not there are infinitely many q with \(K_q \models \phi _{d,\mu }\) can also be done in an effective way [giving a replacement for Theorem 3.5(ii)].

In summary, we have proved the following result, which is Theorem 2 for the Suzuki and Ree families:

3.5 Determining finitely many images

We continue with the notation of Sect. 3.3. In particular \({\mathcal {Y}}\) is the class defined at the start of that section (or, given the remarks in Sect. 3.4, one of the classes of Suzuki or Ree groups). We will denote \(\mathrm{ID}(^d\!X(q))\) by H(q) (this is consistent with the above notation where \(H({\mathbb {F}}_q)\) is the interpretation of \(\mathrm{ID}(^d\!X(q))\) in \({\mathbb {F}}_q)\).

We suppose that \({\mathcal {Q}}(G)\) is finite and give a primitive recursive algorithm to determine it. From the above, we have a finite list of the possible primes dividing the elements of \({\mathcal {Q}}(G)\), so we fix p to be one of these and determine \({\mathcal {Q}}_p(G)\), the images in \({\mathcal {Y}}\) in this characteristic. By Lemma 3.9, we can test whether this is empty or not by applying Theorem 3.5(ii) to the formula \(\lnot (\exists \bar{x}\Phi (\bar{x}))\).

From now on, we assume that \({\mathcal {Q}}_p(G)\) is non-empty.

Let \(E = \{ e : {\mathbb {F}}_{p^e} \models (\exists \bar{x})(\Phi (\bar{x}))\}\). Note that this is infinite. By Lemma 3.11, the d-part of elements of this set is bounded (and we can compute a bound effectively, as in the proof of Lemma 3.11). Thus it will suffice to show that we can effectively determine the elements of \({\mathcal {Q}}_p(G)\) which are defined over fields \({\mathbb {F}}_{p^e}\) where the d-part of e is some fixed power a of d. Call this \({\mathcal {Q}}_p^a(G)\). In what follows, \(q = p^e\) (sometimes with embellishments) will always denote a power of p with d-part a. Note that there is a formula \(\chi _a\) such that \({\mathbb {F}}_{p^f} \models \chi _a\) if and only if f has d-part a.

We shall show how to determine some q such that H(q) contains a copy of each \(T_i\). Note that such a q exists, by Corollary 3.4.

For notational convenience, take \(i = 0\) where i is as in Lemma 3.19. Note that by Lemma 3.19, this i can be determined explicitly (as the constants \(\mu _j\), \(c_j\) are effectively computable). We obtain: