Recognizing finite matrix groups over infinite fields

TL;DR

This paper introduces a comprehensive method for computing with finitely generated matrix groups over infinite fields, solving the finiteness decision problem and enabling structural analysis through finite field isomorphisms.

Contribution

It provides a novel uniform approach for matrix group computations over infinite fields, including finiteness decision and finite field isomorphism construction algorithms.

Findings

Decided finiteness for finitely generated matrix groups over infinite fields

Developed algorithms to construct isomorphic copies over finite fields

Implemented algorithms successfully in MAGMA

Abstract

We present a uniform methodology for computing with finitely generated matrix groups over any infinite field. As one application, we completely solve the problem of deciding finiteness in this class of groups. We also present an algorithm that, given such a finite group as input, in practice successfully constructs an isomorphic copy over a finite field, and uses this copy to investigate the group's structure. Implementations of our algorithms are available in MAGMA.

Equations

$$qb+\left(\genfrac{}{}{0ex}{}{q}{2}\right)b^2+\cdots +b^q=0_n.$$

$$q{b}_{rs}+\left(\genfrac{}{}{0ex}{}{q}{2}\right)b_{rs}^{(2)}+\cdots +b_{rs}^{(q)}=0$$

$$b_{rs}=q{b}_{rs}x+b_{rs}y$$

Full text

1. Introduction

This paper establishes a uniform methodology for computing with finitely generated linear groups over any infinite field. Our techniques constitute a computational analogue of ‘finite approximation’ [24, Chapter 4], which is a major tool in the study of finitely generated linear groups. It relies on the fact that each finitely generated linear group $G$ is residually finite. Moreover, $G$ is approximated by matrix groups of the same degree over finite fields [26, Theorem A, p. 151]. We also use the fundamental result that $G$ has a normal subgroup of finite index with every torsion element unipotent [24, 4.8, p. 56]. For computational purposes, the key objective is to determine a congruence homomorphism whose kernel has this property, and whose image is defined over a finite field.

The first problem that we solve is a natural and obvious candidate for an application of our methodology: testing finiteness of finitely generated linear groups. This problem has been investigated previously, but only for groups over specific domains. Algorithms for testing finiteness over the rational field $\mathbb{Q}$ are given in [2]. One of these, based on integrality testing, is exploited as part of the default procedures in GAP [14] and Magma [3] to decide finiteness over $\mathbb{Q}$. Groups over a characteristic zero function field are considered in [22]. However, the algorithm there possibly involves squaring dimensions. Function fields are also dealt with in [5, 8, 9, 18, 22], where computing in matrix algebras plays a central role. While the algorithms from [8, 9] have been implemented in Magma, we know of no implementations of those from [5, 18, 22].

In this paper, we design a new finiteness testing algorithm that may be employed, for the first time, over any infinite field. The algorithm is concise and practical. Our implementation is distributed with Magma, and we demonstrate that it performs well for a range of inputs.

If a group $G$ is finite then, in practice, we can often construct an isomorphic copy of $G$ over some finite field. As a consequence, drawing on recent progress in computing with matrix groups over finite fields [1, 21], we obtain the first algorithms to answer many structural questions about $G$. These include: computing $|G|$; testing membership in $G$; computing Sylow subgroups, a composition series, and the solvable and unipotent radicals of $G$.

We emphasize that this paper provides a framework for the solution of broader computational problems than the testbed ones treated here. SW-homomorphisms (defined below) are used in [7] to test nilpotency over certain fields. In [11], these are extended to decide virtual properties of finitely generated linear groups. The present paper gives a comprehensive account of our techniques that is valid in all settings. For further discussion of how these ideas have been developed, see the survey [6].

Briefly, the paper is organized as follows. Sections 2 and 3 set up our computational analogue of finite approximation. The algorithms are presented and justified in Section 4. In the final section, we report on our Magma implementation.

2. Congruence homomorphisms of finitely generated linear

groups

Let $\rho$ be a proper ideal of an (associative, unital) ring $\Delta$. The natural surjection $\Delta\rightarrow\Delta/\rho$ induces an algebra homomorphism $\mathrm{Mat}(n,\Delta)\rightarrow\mathrm{Mat}(n,\Delta/\rho)$, which restricts to a group homomorphism $\mathrm{GL}(n,\Delta)\rightarrow\mathrm{GL}(n,\Delta/\rho)$. All these congruence homomorphisms will be denoted by $\phi_{\rho}$. The principal congruence subgroup $\Gamma_{\rho}$ is the kernel of $\phi_{\rho}$ in $\mathrm{GL}(n,\Delta)$.

We fix some more notation, used throughout. Let $S=\{g_{1},\ldots,g_{r}\}\subseteq\mathrm{GL}(n,\mathbb{F})$, where $\mathbb{F}$ is a field. Denote $\langle S\hskip 0.56905pt\rangle$ by $G$. Then $G\leq\mathrm{GL}(n,R)$, where $R\subseteq\mathbb{F}$ is the (Noetherian) ring generated by the entries of the matrices $g_{i}$, $g_{i}^{-1}$, $1\leq i\leq r$. Recall that $R/\rho$ is a finite field if $\rho$ is a maximal ideal (see [24, p. 50]). For the purpose of studying $G$, we may assume without loss of generality that $\mathbb{F}$ is the field of fractions of $R$, and is a finitely generated extension of its prime subfield.

Each finitely generated linear group possesses a normal subgroup $N$ of finite index whose torsion elements are all unipotent; so $N$ is torsion-free if $\mathrm{char}\,R=0$. A proof of this result, due to Selberg (1960) and Wehrfritz (1970), can be found in [24, 4.8, p. 56]; a short new proof is supplied by Proposition 2.1 and Corollary 2.2 below. We call such a normal subgroup $N$ of a given linear group an SW-subgroup. If $\rho$ is an ideal of $R$ such that $\Gamma_{\rho}$ is an SW-subgroup of $\mathrm{GL}(n,R)$, then $\phi_{\rho}$ is an SW-homomorphism. We now formulate conditions that enable us to construct SW-homomorphisms.

Proposition 2.1**.**

Let $\Delta$ be a Noetherian integral domain, and $\rho$ be a maximal ideal of $\Delta$. If $\Gamma_{\rho}$ has a non-trivial torsion element $h$, then $p:=\mathrm{char}(\Delta/\rho)>0$ and $|h|$ is a power of $p$.

Proof.

Set $|h|=q$. Since $\phi_{\rho}(h)=1_{n}$, we have $h=1_{n}+b$ where $b\in\mathrm{Mat}(n,\rho)$, $b\neq 0_{n}$. Hence $(1_{n}+b)^{q}=1_{n}$, so that

[TABLE]

For $k\geq 2$, denote the $(i,j)$th entry of $b^{k}$ by $b_{ij}^{(k)}$; then $b_{ij}^{(k)}\in\rho^{k}$. By the Krull Intersection Theorem [17, 27.8, p. 437], $\cap_{k=1}^{\infty}\rho^{k}=\{0\}$. Hence there exists a positive integer $c$ such that $b_{ij}\in\rho^{c}$ for all $i,j$, but $b_{rs}\notin\rho^{c+1}$ for some $r,s$. This implies that $b_{ij}^{(k)}\in\rho^{kc}$. Now

[TABLE]

by ($\dagger$), so $qb_{rs}\in\rho^{2c}\subseteq\rho^{c+1}$.

Suppose that $q\notin\rho$. Since $\Delta/\rho$ is a field, there exist $x\in\Delta$ and $y\in\rho$ such that $1=qx+y$. Then

[TABLE]

and so, because $b_{rs}\in\rho^{c}$ and $qb_{rs}\in\rho^{c+1}$, we get $b_{rs}\in\rho^{c+1}$. This contradiction proves that $q\in\rho$. Thus $q$ must be a power of $p$. For if not, we could have begun with $h$ of prime order different to $p$, and then $\rho$ would contain two different prime integers and so would contain $1$. ∎

Corollary 2.2**.**

Let $\rho$, $\rho_{1}$, and $\rho_{2}$ be maximal ideals of the Noetherian integral domain $\Delta$.

• (i)

If $\mathrm{char}(\Delta/\rho)=0$ then $\Gamma_{\rho}$ is torsion-free.

• (ii)

Suppose that $\mathrm{char}\,\Delta=0$, and $\mathrm{char}(\Delta/\rho_{1})\neq\mathrm{char}(\Delta/\rho_{2})$. Then $\Gamma_{\rho_{1}}\cap\Gamma_{\rho_{2}}$ is a torsion-free subgroup of $\mathrm{GL}(n,\Delta)$. In particular, if $\Delta=R$ then $\Gamma_{\rho_{1}}\cap\Gamma_{\rho_{2}}$ is an SW-subgroup of $\mathrm{GL}(n,R)$.

• (iii)

Suppose that $\mathrm{char}\,\Delta=p>0$. Then each torsion element of $\Gamma_{\rho}$ is unipotent. In particular, if $\Delta=R$ then $\Gamma_{\rho}$ is an SW-subgroup of $\mathrm{GL}(n,R)$.

Proof.

Clear from Proposition 2.1. ∎

Note that parts (i) and (ii) of Corollary 2.2 contribute to a solution of the problem posed on p. 70 of [23].

By Corollary 2.2 (ii), if $\mathrm{char}\,R=0$ then an SW-subgroup can be constructed as the intersection of two congruence subgroups. Since this may not be convenient, we mention one more result.

Proposition 2.3**.**

Suppose that $\Delta$ is a Dedekind domain of characteristic zero, and $\rho$ is a maximal ideal of $\Delta$ such that $\mathrm{char}(\Delta/\rho)=p>2$. If $p\notin\rho^{2}$ then $\Gamma_{\rho}$ is torsion-free.

Proof.

See [23, Theorem 4, p. 70]. ∎

3. Construction of SW-homomorphisms

We now outline methods to construct both congruence homomorphisms and SW-homomorphisms, given the assumptions on $\mathbb{F}$ made in the second paragraph of Section 2.

Since $\mathbb{F}$ is a finitely generated extension of its prime subfield, there is a subfield $\mathbb{P}\subseteq\mathbb{F}$ of finite degree over the prime subfield, and elements $x_{1},\ldots,x_{m}$ ($m\geq 0$) algebraically independent over $\mathbb{P}$, such that $\mathbb{F}$ is a finite extension of $\mathbb{L}=\mathbb{P}(x_{1},\ldots,x_{m})$; say $|\mathbb{F}:\mathbb{L}|=e\geq 1$. Here $|\mathbb{P}:\mathbb{Q}|=k\geq 1$ if $\mathrm{char}\,\mathbb{F}=0$, and if $\mathrm{char}\,\mathbb{F}=p>0$ then $\mathbb{P}$ is the field $\mathbb{F}_{q}$ of size $q$.

Each type of field is considered in its own section below. For an integral domain $\Delta$ and $\mu\in\Delta\setminus\{0\}$, let $\frac{1}{\mu}\Delta$ denote the ring of fractions with denominators in the multiplicative submonoid of $\Delta$ generated by $\mu$.

3.1. The rational field

Let $\mathbb{F}=\mathbb{Q}$. Then $R=\frac{1}{\mu}\mathbb{Z}$ where $\mu$ is the least common multiple of the denominators of the entries in the matrices $g_{i},g_{i}^{-1}$, $1\leq i\leq r$. For a prime $p\in\mathbb{Z}$ not dividing $\mu$, define $\phi_{1}=\phi_{1,p}:\mathrm{GL}(n,R)\rightarrow\mathrm{GL}(n,p)$ to be entry-wise reduction modulo $p$. If $p>2$ then we denote $\phi_{1,p}$ by $\Phi_{1}=\Phi_{1,p}$. By Proposition 2.3, $\Phi_{1}$ is an SW-homomorphism.

3.2. Number fields

Let $\mathbb{F}$ be a number field, so that $\mathbb{F}=\mathbb{Q}(\alpha)$ for some algebraic number $\alpha$. Let $f(t)$ be the minimal polynomial of $\alpha$, of degree $k$. Multiplying $\alpha$ by a common multiple of the denominators of the coefficients of $f(t)$, if necessary, we may assume that $\alpha$ is an algebraic integer; that is, $f(t)\in\mathbb{Z}[t]$.

We have $R\subseteq\frac{1}{\mu}\mathbb{Z}[\alpha]\subseteq\frac{1}{\mu}\mathcal{O}$ for some $\mu\in\mathbb{Z}$, where $\mathcal{O}$ is the ring of integers of $\mathbb{F}$. We define an SW-homomorphism on $R$ as the restriction of a congruence homomorphism on the Dedekind domain $\frac{1}{\mu}\mathcal{O}$.

Let $p\in\mathbb{Z}$ be a prime not dividing $\mu$, and denote by $\bar{f}(t)$ the polynomial obtained by mod $p$ reduction of the coefficients of $f(t)$. Further, let $\bar{\alpha}$ be a root of $\bar{f}(t)$, so that $\bar{\alpha}$ is a root of some $\mathbb{Z}_{p}$-irreducible factor $\bar{f}_{j}(t)$ of $\bar{f}(t)$. Each $b\in R$ may be expressed uniquely in the form $b=\sum^{k-1}_{i=0}c_{i}\alpha^{i}$ where $c_{i}\in\frac{1}{\mu}\mathbb{Z}$. Thus the assignment $\phi_{2,p}:b\mapsto\sum^{k-1}_{i=0}\phi_{1,p}(c_{i})\bar{\alpha}^{i}$ is well-defined. Moreover, $\phi_{2,p}$ is a ring homomorphism $R\rightarrow\mathbb{Z}_{p}(\bar{\alpha})=\mathbb{F}_{p^{l}}$, say. Thus we have an induced congruence homomorphism $\phi_{2,p}:\mathrm{GL}(n,R)\rightarrow\mathrm{GL}(n,p^{l})$.

Next, we state criteria under which $\phi_{2}=\phi_{2,p}$ is an SW-homomorphism.

Lemma 3.1**.**

Suppose that $p\in\mathbb{Z}$ is an odd prime dividing neither $\mu$ nor the discriminant of $f(t)$. Then the kernel of $\phi_{2,p}$ on $\mathrm{GL}(n,R)$ is torsion-free.

Proof.

Let $f_{j}(t)$ be a preimage of $\bar{f}_{j}(t)$ in $\mathbb{Z}[t]$. The ideal $\rho$ generated by $p$ and $f_{j}(\alpha)$ in $\frac{1}{\mu}\mathcal{O}$ is maximal, by [19, Theorem 3.8.2]. Hence $\rho\cap R$ is a maximal ideal of $R$. Also $p\not\in\rho^{2}$ by [19, Proposition 3.8.1, Theorem 3.8.2]. The lemma then follows from Proposition 2.3. ∎

Lemma 3.2**.**

There are no non-trivial $p$-subgroups of $\mathrm{GL}(n,\mathbb{F})$ if $p>nk+1$.

Proof.

Let $g\in\mathrm{GL}(n,\mathbb{Q})$ be of order $p$. Since the characteristic polynomial of $g$ has a primitive $p$th root of unity as a root, it is divisible by the $p$th cyclotomic polynomial. Thus $p-1\leq n$. The general claim holds because each subgroup of $\mathrm{GL}(n,\mathbb{F})$ is isomorphic to a subgroup of $\mathrm{GL}(nk,\mathbb{Q})$. ∎

Corollary 3.3**.**

Suppose that $\Delta$ is a Noetherian subring of $\mathbb{F}$, and $\rho$ is a maximal ideal of $\Delta$ such that $\mathrm{char}(\Delta/\rho)=p>nk+1$. Then $\Gamma_{\rho}$ is torsion-free.

Proof.

This is a consequence of Proposition 2.1 and Lemma 3.2. ∎

Let $p\in\mathbb{Z}$ be a prime not dividing $\mu$. We denote $\phi_{2,p}$ by $\Phi_{2}=\Phi_{2,p}$ if one of the following extra conditions on $p$ is satisfied: $p$ is odd and does not divide the discriminant of the minimal polynomial of $\alpha$; or $p>nk+1$. The preceding discussion shows that $\Phi_{2}$ is an SW-homomorphism.

Example 3.4**.**

Suppose that $\mathbb{F}$ is a cyclotomic field, say $\mathbb{F}=\mathbb{Q}(\zeta)$ where $\zeta$ is a primitive $c$th root of unity, $c>2$. If $p>2$ and $p$ does not divide $\mathrm{lcm}(\mu,c)$, then $\phi_{2,p}$ is an SW-homomorphism by Lemma 3.1.

3.3. Function fields

Let $\mathbb{F}=\mathbb{P}(x_{1},\ldots,x_{m})$, $m\geq 1$, where $\mathbb{P}$ is $\mathbb{Q}$, a number field, or $\mathbb{F}_{q}$. We have $R\subseteq\frac{1}{\mu}\mathbb{P}[x_{1},\ldots,x_{m}]$ for some $\mu=\mu(x_{1},\ldots,x_{m})$ determined by $S\cup S^{-1}$.

Let $a=(a_{1},\ldots,a_{m})$ be a non-root of $\mu$. If $\mathrm{char}\,\mathbb{F}=0$, then $a_{i}\in\mathbb{P}$ for all $i$; if $\mathbb{F}$ has positive characteristic, then the $a_{i}$ are in $\mathbb{P}$ or some finite extension. Define $\phi_{3}=\phi_{3,a}$ to be the map that substitutes $a_{i}$ for $x_{i}$, $1\leq i\leq m$. Corollary 2.2 (i) implies that $\phi_{3}:\mathrm{GL}(n,R)\rightarrow\mathrm{GL}(n,\mathbb{P})$ is a homomorphism with torsion-free kernel if $\mathrm{char}\,\mathbb{F}=0$. We then obtain an SW-homomorphism in zero characteristic by setting $\Phi_{3}=\Phi_{3,a,p}=\Phi_{i,p}\circ\phi_{3,a}$, where $i=1$ or $2$ if $\mathbb{P}=\mathbb{Q}$ or $\mathbb{P}$ is a number field, respectively. If $\mathbb{P}=\mathbb{F}_{q}$ then $\Phi_{3}=\phi_{3}$ is an SW-homomorphism by Corollary 2.2 (iii). Notice that $\Phi_{3,a,p}$ is defined for all but a finite number of $a$ and $p$ when $m=1$; otherwise, $\Phi_{3,a,p}$ is defined for infinitely many $a$ and $p$.

3.4. Algebraic function fields

For $m\geq 1$, let $\mathbb{L}=\mathbb{P}(x_{1},\ldots,x_{m})$ and $\mathbb{L}_{0}=\mathbb{P}[x_{1},\dots,x_{m}]$, where again $\mathbb{P}$ is $\mathbb{Q}$, a number field, or $\mathbb{F}_{q}$. We assume that $\mathbb{F}=\mathbb{L}(\alpha)$ is a simple extension of $\mathbb{L}$ of degree $e>1$. For instance, we can stipulate that $\mathbb{F}$ is a separable extension of $\mathbb{L}$ (e.g., in characteristic $p$ this is assured if $p\nmid e$). Let $f(t)\in\mathbb{L}_{0}[t]$ be the minimal polynomial of $\alpha$. We have $R\subseteq\frac{1}{\mu}\mathbb{L}_{0}[\alpha]$ for some $\mu\in\mathbb{L}_{0}$ determined in the usual way by the input $S$.

Suppose that $a=(a_{1},\ldots,a_{m})$ is a non-root of $\mu$, where the $a_{i}$ are in $\mathbb{P}$ or a finite extension. Denote by $\bar{f}(t)$ the polynomial obtained by substitution of $a$ in the coefficients of $f(t)$. Define $\bar{c}=\phi_{3,a}(c)$ for $c\in\frac{1}{\mu}\mathbb{L}_{0}$ similarly. Let $\bar{\alpha}$ be a root of $\bar{f}(t)$. Define $\phi_{4}=\phi_{4,a}:R\rightarrow\mathbb{P}(\bar{\alpha})$ by $\phi_{4}:\sum_{i=0}^{e-1}c_{i}\alpha^{i}\mapsto\sum_{i=0}^{e-1}\bar{c}_{i}\bar{\alpha}^{i}$. Therefore, if $\rm{char}\,\mathbb{F}=0$ then we get an induced congruence homomorphism $\phi_{4}:\mathrm{GL}(n,R)\rightarrow\mathrm{GL}(n,\mathbb{P}(\bar{\alpha}))$, whose kernel is torsion-free by Corollary 2.2 (i). Set $\Phi_{4}=\Phi_{4,a,p}=\Phi_{i,p}\circ\phi_{4,a}$, where $i=1$ if $\mathbb{P}(\bar{\alpha})=\mathbb{Q}$, and $i=2$ if $\mathbb{P}(\bar{\alpha})$ is a number field. If $\mathrm{char}\,\mathbb{F}>0$ then we set $\Phi_{4}=\phi_{4}$. In all cases $\Phi_{4}$ is an SW-homomorphism. As with $\Phi_{3,a,p}$, the homomorphism $\Phi_{4,a,p}$ is defined for infinitely many $a$ and $p$, and for all but a finite number of $a$, $p$ when $m=1$.

Remark 3.5*.*

Fields $\mathbb{F}$ as in Sections 3.1–3.4 are the main ones supported by GAP and Magma.

Remark 3.6*.*

SW-homomorphisms are used in [11, Section 5.3] to test whether $G\leq\mathrm{GL}(n,\mathbb{F})$ is central-by-finite; indeed, each ‘W-homomorphism’ defined in that paper is a special kind of SW-homomorphism. They also feature in the nilpotency testing algorithm of [7].

3.5. Analyzing congruence homomorphisms

We now prove some results that will be helpful in the analysis of our algorithms.

Lemma 3.7**.**

Let $\Delta$ be a Dedekind domain, and let $G$ be a finitely generated subgroup of $\mathrm{GL}(n,\Delta)$. For all but a finite number of maximal ideals $\rho$ of $\Delta$, the following are true:

• (i)

if $G$ is finite then $\phi_{\rho}$ is an isomorphism of $G$ onto $\phi_{\rho}(G)$;

• (ii)

if $G$ is infinite, and $\nu$ is a positive integer, then $\phi_{\rho}(G)$ contains an element of order greater than $\nu$.

Proof.

(Cf. [24, p. 51] and [8, Lemma 3].) Note that a non-zero element $a$ of $\Delta$ is contained in only finitely many maximal ideals of $\Delta$. To see this, let $a\Delta=\rho_{1}^{e_{1}}\cdots\rho_{c}^{e_{c}}$, where the $\rho_{i}$ are maximal ideals. If $\rho$ is a maximal ideal of $\Delta$ containing $a$, then $\rho_{1}^{e_{1}}\cdots\rho_{c}^{e_{c}}\subseteq\rho$, so $\rho=\rho_{i}$ for some $i$.

Next, let $M=\{h_{1},\ldots,h_{d}\}\subseteq\mathrm{Mat}(n,\Delta)$, and for each pair $l,k\in\{1,\ldots,d\}$, $l\neq k$, choose $(i,j)$ such that $h_{l}(i,j)-h_{k}(i,j)\neq 0$. Denote the product of all differences $h_{l}(i,j)-h_{k}(i,j)$ by $a_{M}$. If $\rho$ is an ideal of $\Delta$ not containing $a_{M}$, then $|\phi_{\rho}(M)|=|M|$.

Taking $M$ to be the set of elements of $G$, part (i) is now clear.

If $G$ is infinite then $G$ contains an element $g$ of infinite order, by a result of Schur [23, Theorem 5, p. 181]. Thus, taking $M$ to be $\{g,\ldots,g^{\nu},g^{\nu+1}\}$, we get (ii). ∎

To utilize Lemma 3.7 in our context, let $\mathbb{F}$ be one of $\mathbb{Q}$, a number field, $\mathbb{P}(x)$, or a finite extension of $\mathbb{P}(x)$. The relevant SW-homomorphism $\Phi$ on $\mathrm{GL}(n,R)$ is the restriction of a congruence homomorphism $\phi_{\rho}$ on $\mathrm{GL}(n,\Delta)$, where $\Delta$ is a Dedekind domain with maximal ideal $\rho$. Hence for $G\leq\mathrm{GL}(n,R)$ and all but a finite number of choices in the definition of $\phi_{\rho}$, the following hold: (a) if $G$ is finite, then $\Phi$ is an isomorphism on $G$; (b) if $G$ is infinite, then $\Phi(G)$ contains an element of order greater than any given positive integer $\nu$. For the other fields $\mathbb{F}$ where $R$ may not be contained in a Dedekind domain (function fields with more than one indeterminate, or finite extensions thereof), it is still true that there are infinitely many SW-homomorphisms $\Phi$ such that (a) and (b) hold. This follows from the definition of $\Phi$ in each case, and arguing as in the proof of Lemma 3.7.

4. Finiteness algorithms for matrix groups

4.1. Preliminaries: asymptotic bounds

We continue with the notation of the previous section: $|\mathbb{F}:\mathbb{L}|=e\geq 1$, $\mathbb{L}=\mathbb{P}(x_{1},\ldots,x_{m})$, $m\geq 0$, and $|\mathbb{P}:\mathbb{Q}|=k\geq 1$ or $\mathbb{P}=\mathbb{F}_{q}$.

Suppose first that $\mathrm{char}\,\mathbb{F}=0$. Put $n_{0}=nke$.

Lemma 4.1**.**

A finite subgroup $G$ of $\mathrm{GL}(n,\mathbb{F})$ is isomorphic to a subgroup of $\mathrm{GL}(n_{0},\mathbb{Q})$.

Proof.

Certainly $G$ is isomorphic to a subgroup of $\mathrm{GL}(ne,\mathbb{L})$, and a subgroup of $\mathrm{GL}(ne,\mathbb{P})$ is isomorphic to a subgroup of $\mathrm{GL}(nke,\mathbb{Q})$. The lemma follows from [23, p. 69, Corollary 4]. ∎

It is well-known that the order of a finite subgroup of $\mathrm{GL}(n,\mathbb{Q})$ is bounded by a function of $n$ (see, e.g., [12, 13]). Hence by Lemma 4.1 there are functions $\nu_{1}=\nu_{1}(n_{0})$ and $\nu_{2}=\nu_{2}(n_{0})$ bounding the order of a finite subgroup of $\mathrm{GL}(n,\mathbb{F})$ and the order of a torsion element of $\mathrm{GL}(n,\mathbb{F})$, respectively. For $n_{0}>10$ or $n_{0}=3,5$ we may take $\nu_{1}=2^{n_{0}}(n_{0})!$ by [12, Theorem A]; for the remaining $n_{0}$, values of $\nu_{1}$ are also listed there. A suitable function $\nu_{2}$ is given by the next lemma.

Lemma 4.2**.**

If $g$ is a torsion element of $\mathrm{GL}(n,\mathbb{F})$ then $|g|\leq 2^{\lfloor\log_{2}n_{0}\rfloor+1}3^{\left\lfloor n_{0}/2\right\rfloor}$.

Proof.

Let $\mathbb{F}=\mathbb{Q}$. If $|g|$ is odd then $|g|\leq 3^{\left\lfloor n/2\right\rfloor}$ by [13, p. 3519]. Suppose that $g$ is a $2$-element. Then $g$ is conjugate to a monomial matrix over $\mathbb{Q}$ (see [20, IV.4]). Since the order of a $2$-element in $\mathrm{Sym}(n)$ is bounded by the largest power $2^{t}$ of $2$ less than or equal to $n$, $|g|\leq 2^{t+1}$. Lemma 4.1 now implies the result in the general case $\mathbb{F}\supseteq\mathbb{Q}$. ∎

Here is one more useful condition to detect infinite groups in characteristic zero.

Lemma 4.3**.**

If $G\leq\mathrm{GL}(n,\mathbb{F})$ is finite and $p>n_{0}+1$ then $p\nmid|G|$.

Proof.

This follows from Lemmas 3.2 and 4.1. ∎

Now suppose that $\mathrm{char}\,\mathbb{F}>0$. The order of a finite subgroup of $\mathrm{GL}(n,\mathbb{F})$ can be arbitrarily large. On the other hand, the orders of torsion elements of $\mathrm{GL}(n,\mathbb{F})$ are bounded. The next lemma furnishes such a bound.

Lemma 4.4**.**

Let $n_{0}=ne$. If $g$ is a torsion element of $\mathrm{GL}(n,\mathbb{F})$ then $|g|\leq q^{n_{0}}-1$.

Proof.

The proof is essentially the same as that of [22, Theorem 3.3, Corollary 3.4]. We recap the main points. It suffices to assume that $\mathbb{F}=\mathbb{L}$. By [25], $g$ is conjugate to a block upper triangular matrix, where the (irreducible) blocks are $\mathbb{F}_{q}$-matrices. Hence the characteristic polynomial of $g$ has $\mathbb{F}_{q}$-coefficients. It follows that the dimension of $\langle\hskip 0.56905ptg\hskip 0.42677pt\rangle_{\mathbb{F}_{q}}$ is at most $n$, and so every invertible element of this enveloping algebra has order at most $q^{n}-1$. ∎

4.2. Testing finiteness

By Section 3, we are able to construct a congruence image $\phi_{\rho}(G)$ of $G\leq$ $\mathrm{GL}(n,\mathbb{F})$ over a finite field such that the torsion elements of $G_{\rho}:=G\cap\Gamma_{\rho}$ are unipotent. Thus, to decide finiteness of $G$, we merely test whether $G_{\rho}$ is trivial ($\mathrm{char}\,\mathbb{F}=0$), or whether $G_{\rho}$ is unipotent ($\rm{char}\,\mathbb{F}>0$). Both tasks can be accomplished with just normal generators of $G_{\rho}$: generators for a subgroup whose normal closure in $G$ is $G_{\rho}$. That is, we do not need to construct the full congruence subgroup. Normal generators are found by a standard method [16, pp. 299–300] that requires a presentation of $\phi_{\rho}(G)$ as input. Since it is a matrix group over a finite field, we can compute a presentation of $\phi_{\rho}(G)$ using the algorithms described in [1, 21]. We refer to such an algorithm as $\tt Presentation$. Let $\tt SWImage$ be an algorithm that constructs a congruence image over a finite field. The congruence homomorphism in question is one of the SW-homomorphisms $\Phi=\Phi_{i}$, $1\leq i\leq 4$, defined in Sections 3.1–3.4. The following procedure tests finiteness along the lines just explained (see Section 4.1 for definitions of $n_{0}$ and $\nu_{1}$).

$\tt IsFiniteMatrixGroup$

Input: $S=\{g_{1},\ldots,g_{r}\}\subseteq\mathrm{GL}(n,\mathbb{F})$.

Output: $\tt true$ if $G=\langle\hskip 0.56905ptS\,\hskip 0.42677pt\rangle$ is finite; $\tt false$ otherwise.

1. (1)

$H:={\tt SWImage}(G)=\langle\hskip 0.56905pt\Phi(g_{1}),\ldots,\Phi(g_{r})\hskip 0.42677pt\rangle$. 2. (2)

If $\mathrm{char}\,\mathbb{F}=0$ and either $|H|>\nu_{1}$ or $p$ divides $|H|$ for some prime $p>n_{0}+1$, then return $\tt false$. 3. (3)

${\tt Presentation}(H):=\langle\hskip 0.78242pt\Phi(g_{1}),\ldots,\Phi(g_{r})\hskip 2.84526pt|\hskip 3.55658pt\omega_{j}(\Phi(g_{1}),\ldots,\Phi(g_{r}))=1;1\leq j\leq t\hskip 0.7113pt\rangle$. 4. (4)

$K:=\{\omega_{j}(g_{1},\ldots,g_{r})\mid 1\leq j\leq t\}$. 5. (5)

If $\mathrm{char}\,\mathbb{F}=0$ and $K=\{1_{n}\}$, or $\mathrm{char}\,\mathbb{F}>0$ and ${\tt IsUnipotent}(\langle\hskip 0.56905ptK\hskip 0.42677pt\rangle^{G})$, then return $\tt true$. Else return $\tt false$.

Step (2) is justified by Lemma 4.3 and the comments before Lemma 4.2. For example, if $\mathbb{F}$ is a number field then Lemma 3.7 suggests that the initial check in this step will usually identify that $G$ is infinite. We test unipotency of the congruence subgroup $\langle\hskip 0.56905ptK\hskip 0.42677pt\rangle^{G}$ in step (5) using the normal generating set $K$. A procedure for doing this, based on computation in enveloping algebras, is given in [11, Section 5.2]. Also note that we can apply a conjugation isomorphism as in [15] to write the SW-image over the smallest possible finite field of the chosen characteristic.

Next we consider the special but very important case that $G$ is a cyclic group: testing whether $g\in\mathrm{GL}(n,\mathbb{F})$ has finite order. Let $\nu_{2}$ be an upper bound on the order of a torsion element of $\mathrm{GL}(n,\mathbb{F})$. See Lemmas 4.2 and 4.4 for values of $\nu_{2}$.

$\tt IsFiniteCyclicMatrixGroup$

Input: $g\in\mathrm{GL}(n,\mathbb{F})$.

Output: $\tt true$ if $g$ has finite order; $\tt false$ otherwise.

1. (1)

$h:={\tt SWImage}(g)$. 2. (2)

$d:={\tt Order}(h)$. 3. (3)

If $d>\nu_{2}$, or $\mathrm{char}\,\mathbb{F}=0$ and $p\mid d$ for some prime $p>n_{0}+1$, then return $\tt false$. 4. (4)

If $\mathrm{char}\,\mathbb{F}=0$ and $g^{d}=1_{n}$, or $\mathrm{char}\,\mathbb{F}>0$ and ${\tt IsUnipotent}(g^{d})$, then return $\tt true$. Else return $\tt false$.

Note that $g^{d}$ is unipotent in characteristic $p>0$ if and only if its order divides $p^{\left\lceil\log_{p}n\right\rceil}$ (see [23, p. 192]). Also, if $\mathrm{char}\,\mathbb{F}=0$ and $\tt IsFiniteCyclicMatrixGroup$ returns $\tt true$, then the order $d$ of $g$ is calculated in step (2). In the situations covered by Lemma 3.7, if $|g|$ is infinite then $d>\nu_{2}$ for all but a finite number of choices of $\Phi$. That is, we expect that infiniteness of $|g|$ will be detected at step (3) of $\tt IsFiniteCyclicMatrixGroup$.

Recall that an infinite group $G\leq\mathrm{GL}(n,\mathbb{F})$ has an infinite order element. Hence, as a precursor to running $\tt IsFiniteMatrixGroup$, we check via $\tt IsFiniteCyclicMatrixGroup$ whether ‘random’ elements of $G$, produced by a variation of the product replacement algorithm [4], have infinite order; cf. [2, Section 8.2].

4.3. Recognizing finite matrix groups

Suppose that $G\leq\mathrm{GL}(n,\mathbb{F})$ is finite. We describe how to find an isomorphic copy of $G$ in some $\mathrm{GL}(n,q)$ and carry out further computations with $G$.

If $\mathrm{char}\,\mathbb{F}=0$ then ${\tt SWImage}(G)=\Phi(G)$ is isomorphic to $G$. If $\mathrm{char}\,\mathbb{F}>0$ then the congruence subgroup may be non-trivial. We repeat the construction of normal generators of the congruence subgroup for different choices of $\Phi$, until we find a $\Phi$ for which all these generators are trivial. By the discussion at the end of Section 3.5, if $m=1$ (there is just one indeterminate) then in a finite number of iterations we will get an isomorphic copy of $G$ by Lemma 3.7. Otherwise, although there are infinitely many isomorphisms $\Phi$, the procedure may not terminate. In our many experiments the procedure always succeeded in finding an isomorphic copy of $G$.

Once we have an isomorphic copy, algorithms for matrix groups over finite fields (see [1] and [16, Chapter 10]) are used to investigate the structure and properties of $G$. In particular, we can

• •

compute a composition series and short presentation for $G$;

• •

compute $|G|$;

• •

compute the solvable and unipotent radicals, the derived subgroup, center, and Sylow subgroups of $G$;

• •

test membership of $x\in\mathrm{GL}(n,\mathbb{F})$ in $G$.

Where feasible, the computation is undertaken directly in the isomorphic copy, and the result is ‘lifted’ by means of the known isomorphism to $G$. Sometimes this involves additional work. For instance, membership testing requires that we construct a new isomorphic copy; namely, of $\langle\hskip 0.56905ptG,x\hskip 0.42677pt\rangle$.

5. Implementation and performance

The algorithms have been implemented in Magma as part of our package Infinite [10]. We use machinery from the CompositionTree package [1, 21] to study congruence images and construct their presentations.

We implemented SW-homomorphisms as per Sections 3.1–3.4. These are applied in Infinite to solve specific problems, such as testing finiteness, virtual properties, and nilpotency (the latter over an arbitrary field, significantly enhancing [7]). Here we report on the algorithms of Sections 4.2 and 4.3.

In our implementation of $\tt IsFiniteMatrixGroup$ and $\tt IsFiniteCyclicMatrixGroup$, we construct (at least) two SW-homomorphisms and determine the orders of the images of $G$ under these. If $G$ is finite and $\mathrm{char}\,\mathbb{F}=0$, then the orders must be identical. In positive characteristic, the least common multiple of the orders of two images of an element of finite $G$ must be at most $\nu_{2}$. The single most expensive task is evaluating relations to obtain normal generators for the kernel of an SW-homomorphism, since this may lead to blow-up in the size of matrix entries. Hence we first check the orders of images under several SW-homomorphisms before we evaluate relations.

In [9] we proposed an alternative algorithm to decide finiteness for groups defined over function fields of positive characteristic. This is an option in Infinite; it avoids evaluation of relations over the field of definition, and is sometimes faster than $\tt IsFiniteMatrixGroup$ for such groups.

We now describe sample outputs that illustrate the efficiency and scope of our implementation. The examples chosen cover the main domains and a variety of groups. Our experiments were performed using Magma V2.17-2 on a 2GHz machine. All examples are randomly conjugated, so that generators are not sparse, and matrix entries (numerators and denominators) are large. Since random selection plays a role in some of the CompositionTree algorithms, times stated are averages over three runs. The complete examples are available in the Infinite package.

1. (1)

$G_{1}\leq\mathrm{GL}(24,\mathbb{Q}(\zeta_{17}))$ is a conjugate of the monomial group $\langle\hskip 0.56905pt\zeta_{17}\hskip 0.42677pt\rangle\wr\mathrm{Sym}(24)$. It has order $17^{24}24!$, the maximum possible for a finite subgroup of $\mathrm{GL}(24,\mathbb{Q}(\zeta_{17}))$ by [12]. We decide finiteness of this $3$-generator group and determine its order in $1435$s; compute a Sylow $3$-subgroup in $22$s; and the derived group in $57$s. 2. (2)

$G_{2}\leq\mathrm{GL}(12,\mathbb{F})$ where $\mathbb{F}=\mathbb{P}(x)$ and $\mathbb{P}=\mathbb{Q}(\sqrt{2})$. It is conjugate to $H_{1}\wr H_{2}$ where $H_{1}$ is ${\tt RationalMatrixGroup}(4,2)$ and $H_{2}={\tt PrimitiveSubgroup}(3,1)$, both from standard Magma databases. We decide finiteness of this $7$-generator group in $18$s; compute its order $2^{16}3^{7}$ in $1435$s; its centre in $3$s; and its Fitting subgroup in $3$s. 3. (3)

$G_{3}\leq\mathrm{GL}(20,\mathbb{F})$ where $\mathbb{F}$ is a degree $2$ extension of the function field $\mathbb{Q}(x)$. It is conjugate to the derived subgroup of the monomial group $\langle\hskip 0.56905pt-1\hskip 0.42677pt\rangle\wr\mathrm{Sym}(20)$ in $\mathrm{GL}(20,\mathbb{F})$. We decide finiteness and compute the order of this $31$-generator group in $1090$s; and construct a Sylow $7$-subgroup in $5$s. 4. (4)

$G_{4}\leq\mathrm{GL}(100,\mathbb{Q}(\zeta_{19}))$. We prove that this $14$-generator group is infinite in $9$s. 5. (5)

$G_{5}\leq\mathrm{GL}(30,\mathbb{F})$ where $\mathbb{F}$ is an algebraic function field of degree $3$ over $\mathbb{Q}(x)$. We prove that this $4$-generator group is infinite in $1024$s. 6. (6)

$G_{6}\leq\mathrm{GL}(6,\mathbb{F})$ where $\mathbb{F}$ is an algebraic function field of degree $2$ over $\mathbb{F}_{9}(x)$. It is conjugate to $\mathrm{GL}(6,3^{2})$. We find the order of this $2$-generator group in $18$s; its unipotent radical in $15$s; a Sylow $3$-subgroup $H$ in $18$s; and compute the normalizer in $G_{6}$ of $H$ in $42$s. 7. (7)

$G_{7}\leq\mathrm{GL}(16,\mathbb{F})$ where $\mathbb{F}$ is a degree $3$ extension of $\mathbb{F}_{2}(x)$. It is conjugate to the Kronecker product of $\mathrm{GL}(8,2)$ with a unipotent subgroup of $\mathrm{GL}(2,\mathbb{F}_{2}(x))$. We decide finiteness of this 8-generator group in $16$s; we compute its order $16\cdot|\mathrm{GL}(8,2)|$ and an isomorphic copy in $488$s; and determine the Fitting subgroup in $12$s. 8. (8)

$G_{8}\leq\mathrm{GL}(12,\mathbb{F})$ where $\mathbb{F}$ is a function field with two indeterminates over $\mathbb{F}_{5}$. We prove that this $8$-generator group is infinite in $6$s. 9. (9)

$G_{9}\leq\mathrm{GL}(12,\mathbb{F})$ where $\mathbb{F}$ is a degree $2$ extension of a univariate function field over $\mathbb{F}_{5}$. We prove that this $8$-generator group is infinite in $10$s.