Elsevier

Journal of Algebra

Volume 561, 1 November 2020, Pages 215-236
Journal of Algebra

Constructing composition factors for a linear group in polynomial time

https://doi.org/10.1016/j.jalgebra.2020.02.018Get rights and content

Abstract

We present a Las Vegas polynomial-time algorithm that takes as input a subgroup of GL(d,Fq) and, subject to the existence of certain oracles, determines its composition factors, provided that none of those factors is isomorphic to one of B22(22k+1), F42(22k+1), D43(2k), or G22(32k+1), for any k.

Introduction

In 1987 Luks [41] provided the first polynomial-time algorithm to construct the composition factors of a permutation group. This result has important implications: Kantor [33] employed it to obtain polynomial-time construction of Sylow subgroups, and Babai, Luks & Seress [7] used it as a building block for a family of polynomial-time algorithms for permutation groups. For an extensive related discussion, see Seress [45, §6.2].

Our goal in this paper is to provide the first polynomial-time algorithm to solve this problem for linear groups defined over finite fields. In effect, the algorithm is an outcome of the “matrix group recognition” project, a major topic of research over the past 25 years. For an overview of the project, see [43].

Let G=XGL(d,Fq) where Fq is a finite field of order q=pe. In summary, the fundamental aim of the project is to identify the composition factors of G, and to solve the constructive membership problem in G. This means to decide whether a given gGL(d,Fq) lies in G and, if so, to write g as a word over X: namely, as a word in the alphabet XX1. In practice, we construct a compressed version of the word as a straight line program [45, p. 10]; this ensures that its length (and cost of evaluation) is polynomially bounded.

Two approaches have dominated the research undertaken. Babai & Beals [8] initiated the black-box approach: it aims to construct a specific characteristic series of subgroups for an arbitrary finite group G that can be refined to provide a composition series; the associated algorithms are independent of the given representation of G. In 2009, Babai, Beals & Seress [11] proved that, subject to the availability of certain oracles, there exists a Monte Carlo polynomial-time black-box algorithm to construct this characteristic series for G=XGL(d,Fq) when q is odd, to identify the composition factors, and to solve the constructive membership problem for G. (For the definitions of Monte Carlo and Las Vegas algorithms, see Section 2.1 below.) If q is even, then they can construct a composition series for G/Rad(G) and identify its composition factors, where Rad(G) is the soluble radical of G. Their computations in the soluble radical rely on the work of Luks [42].

The algorithms of [11] rely on two number-theoretic oracles. The first is a discrete log oracle: for given nonzero μ and fixed primitive element ω of a finite field F, it returns the unique k{0,,|F|1} such that μ=ωk. It is needed for fields of order qi for 1id. The second oracle factorises numbers of the form qi1 for 1id. Both are needed to solve problems in abelian matrix groups. The algorithms can be upgraded to Las Vegas provided that polynomial-time black-box constructive membership algorithms and short presentations (both defined below) are available for all nonabelian composition factors of G.

By contrast, the geometric approach investigates whether G=XGL(d,Fq) satisfies certain geometric properties in its action on its underlying vector space V=Fqd. For example, G acts reducibly if it fixes a nonzero proper subspace of V, and it acts imprimitively if it permutes the summands of a direct sum decomposition of V. A classification of the maximal subgroups of classical groups by Aschbacher [1] underpins this approach: in summary, either G preserves a linear structure in its action on V, and has a normal subgroup related to this structure, so providing a reduction; or it has a normal absolutely irreducible subgroup that is simple modulo scalars. The associated algorithms recursively exploit this reduction to construct a composition series for G. The outcome is reported in [4], where the algorithm

is described. It takes as input G=XGL(d,Fq) and outputs a composition tree, a data structure, for G. The tree allows us to list both a composition and chief series for G, and to solve membership and other problems for G.

Central to the

algorithm are short presentations for the simple groups that occur as composition factors of G. For each finite nonabelian simple group S, we have defined a specific sequence of standard generators. A constructive recognition algorithm for S takes as input a group G=X known to be isomorphic to S, computes standard generators of G as words over X, and uses the standard generators to establish an isomorphism between G and (a central quotient of) the standard copy of S, a specific representation of S. The isomorphism is realised by an algorithm that solves the constructive membership problem in G. The constructive recognition algorithm returns the standard generators and the constructive membership algorithm for G. Babai & Szemerédi [5] defined the length of a presentation to be the number of symbols required to write it down. A presentation on our standard generators for every finite nonabelian simple group S is known; with the exception of one family of finite simple groups, this presentation is short in the sense that its length is bounded by a function which is polynomial in log|S|; it is not known whether short presentations exist for the small Ree groups G22(32k+1). For details of the standard generators and presentations, see [15], [17], [38], [40], [48]. Ultimately, these presentations for the composition factors of G are combined to write down a presentation for G, allowing us to verify the correctness of the output of the resulting Las Vegas
algorithm. The outcome is efficient in practice; an implementation of
and its associated algorithms is available in Magma [14].

In the introduction to [4], we wrote that “Serious obstructions remain before we have a provably polynomial-time algorithm to compute a composition tree”. Here we revisit the topic and obtain the following result.

Theorem 1.1

There is a Las Vegas polynomial-time algorithm that takes as input a group G=XGL(d,Fq) and, subject to the existence of a discrete log oracle for Fqi and an oracle to factorise integers of the form qi1 for 1id, and to the availability of polynomial-time constructive recognition algorithms and short presentations for the nonabelian composition factors of G, it constructs a composition tree for G.

By “constructs a composition tree for G”, we mean solving the basic problems discussed earlier: compute a composition series for G, identify the factors in this series, and provide a solution to the constructive membership problem in G. We also provide an isomorphism between each nonabelian composition factor of G and (a central quotient of) the standard copy of that factor.

The following corollary reflects the current status of constructive recognition algorithms for the various families of finite simple groups.

Corollary 1.2

There is a Las Vegas polynomial-time algorithm that takes as input a group G=XGL(d,Fq) that has no composition factor isomorphic to B22(22k+1), F42(22k+1), D43(2k), or G22(32k+1), for any k, and, subject to the existence of a discrete log oracle for Fqi and an oracle to factorise integers of the form qi1 for 1id, it constructs a composition tree for G.

As we shall explain in Section 2.3, the oracle to factorise integers of the form qi1 allows us to calculate and factorise the orders of elements of G in polynomial time. Corollary 1.2 is a direct consequence of Theorem 5.1, which is proved using Theorem 3.1, Theorem 4.1. Although the arguments used in the proofs of these theorems constitute a proof of Theorem 1.1, we preferred to formulate them so that they provide more information on what we can do in the cases excluded by the corollary.

In particular, we can handle groups having G22(32k+1) as composition factors but, since no short presentations are known for the groups G22(32k+1), our algorithm is only Monte Carlo. We can handle individual groups from the other excluded classes for small k by treating them as “sporadic groups”.

The serious obstructions alluded to in [4] to a polynomial-time algorithm arose principally from our inability to find (or prove the non-existence of) Aschbacher reductions of matrix groups in polynomial time. We overcome that problem by proving in Theorem 3.1 that we can in Monte Carlo polynomial time find a nontrivial element in a proper normal subgroup of a nonabelian black-box group, and then prove in Theorem 4.1 that we can use such elements effectively to find Aschbacher reductions of matrix groups. There have also been significant recent advances in the development of algorithms for the constructive recognition of the finite exceptional groups of Lie type.

Our primary objective is to prove the theorem and corollary as stated, without considering the degrees of the polynomials involved. It is easy to produce explicit bounds, but they are too large to be of practical interest. Our implementation of the algorithm of [4] rarely exhibits the difficulties that the algorithm presented here is designed to avoid; this justifies our decision to pay little attention to practical performance.

The discrete log oracle is used in the constructive recognition of simple groups of Lie type, and to determine the order and structure of certain abelian subgroups of GL(d,Fq) that may be rewritten over an extension field Fqi for some i{1,,d}. The most efficient existing algorithms to solve the discrete log problem run in sub-exponential time (see [46, Chapter 4]).

A complete or partial factorisation of integers of the form qi1 for certain i{1,,d} is needed. A partial factorisation into ‘small’ primes and certain coprime residues that are products of ‘large’ primes can be carried out in polynomial time. Further factorisation is only needed if G has a composition factor of order a prime dividing such a residue. That a residue is prime may be determined in polynomial time.

In Section 2 we discuss black-box groups, Monte Carlo and Las Vegas algorithms, and procedures to generate random elements of black-box groups. We also summarise the current status of constructive recognition algorithms for the finite simple groups. In Section 3 we prove the main technical result of the paper by presenting a Monte Carlo algorithm that takes as input a nonabelian black-box group G, and either identifies G as a finite simple group, or outputs a nontrivial element of a proper normal subgroup of G. In Section 4 we show how this algorithm underpins a Las Vegas polynomial-time algorithm that takes as input GGL(d,Fq) and either finds an Aschbacher reduction of G, or proves that G is nearly simple and identifies its nonabelian composition factor. We use this to prove Theorem 1.1 in Section 5.

Section snippets

Black-box groups and algorithms

The concept of a black-box group was introduced in [5]. In this model, the elements of a finite group G are encoded by bit-strings of uniform length N, so G has an encoding of length N and |G|2N. The encoding of an element is not required to be unique, but distinct group elements have distinct encodings. Not all bit-strings are required to represent group elements.

Three oracles are supplied. One takes as input encodings of an ordered pair (g,h) of elements of G and returns an encoding of gh; a

Deciding simplicity

Theorem 3.1

There is a Monte Carlo polynomial-time BBod+ algorithm

that takes as input a nonabelian black-box group G=X and ϵ(0,1/2) and outputs one of the following:
  • (i)

    , the name of G, and a constructive membership algorithm for G;

  • (ii)

    and wG;

  • (iii)

    Fail, possibly with the report that G may have one of the composition factors excluded by condition (b) below.

This output is deemed to be correct if one of the following holds:

  • (1)

    G is simple,

    is returned, the correct name for G is

Deciding reductions for matrix groups

Aschbacher [1] showed that maximal subgroups of classical groups over finite fields are in one of nine classes, which he called C1C8 and S. We extend (or abuse) this notation by applying it to arbitrary subgroups of maximal subgroups in classes C1C7. For example, we view every reducible matrix group as a member of class C1. Viewed in this way, the main result of [1] is that every subgroup G of GL(d,Fq) either lies in at least one of C1C7, or it lies in C8 or S, in which case G has a

A polynomial-time version of

We summarise a mildly simplified version of the

algorithm presented in [4, §3.1]. It takes as input G=XGL(d,Fq) and outputs a composition tree for G.
  • (1)

    Do one of the following:

    • (i)

      construct an effective epimorphism θ:GG1, for some group G1; or

    • (ii)

      prove that G is simple, in which case G becomes a leaf in the tree.

    In Case (i), θ must be a reduction: namely, G1 is “smaller” than G in some respect – for example, its degree or field of definition. Assume henceforth that Case (i) applies.

  • (2)

References (48)

  • William M. Kantor et al.

    Black box groups isomorphic to PGL(2,2e)

    J. Algebra

    (2015)
  • W.M. Kantor et al.

    Black box exceptional groups of Lie type II

    J. Algebra

    (2015)
  • Vicente Landazuri et al.

    On the minimal degrees of projective representations of the finite Chevalley groups

    J. Algebra

    (1974)
  • C.R. Leedham-Green et al.

    Recognising tensor-induced matrix groups

    J. Algebra

    (2002)
  • C.R. Leedham-Green et al.

    Presentations on standard generators for classical groups

    J. Algebra

    (2020)
  • M. Aschbacher

    On the maximal subgroups of the finite classical groups

    Invent. Math.

    (1984)
  • Henrik Bäärnhielm

    Algorithmic problems in twisted groups of Lie type

    (2007)
  • László Babai et al.

    On the complexity of matrix group problems, I

  • László Babai

    Local expansion of vertex-transitive graphs and random generation in finite groups

  • László Babai et al.

    Fast management of permutation groups. I

    SIAM J. Comput.

    (1997)
  • L. Babai et al.

    A polynomial-time theory of black box groups. I

  • L. Babai et al.

    Recognizing simplicity of black-box groups and the frequency of p-singular elements in affine groups

  • L. Babai et al.

    Black-box recognition of finite simple groups of Lie type by statistics of element orders

    J. Group Theory

    (2002)
  • L. Babai et al.

    Polynomial-time theory of matrix groups

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    O'Brien was supported by a University of Auckland Hood Fellowship and by the Hausdorff Institute, Bonn, while this work was carried out. All authors were supported by the Marsden Fund of New Zealand via grant UOA 1626. We thank both James Wilson and the referee for many helpful suggestions.

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