Constructing composition factors for a linear group in polynomial time☆
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 where is a finite field of order . 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 lies in G and, if so, to write g as a word over X: namely, as a word in the alphabet . 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 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 and identify its composition factors, where 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 such that . It is needed for fields of order for . The second oracle factorises numbers of the form for . 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 satisfies certain geometric properties in its action on its underlying vector space . 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 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 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 ; it is not known whether short presentations exist for the small Ree groups . 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 and, subject to the existence of a discrete log oracle for and an oracle to factorise integers of the form for , 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 that has no composition factor isomorphic to , , , or , for any k, and, subject to the existence of a discrete log oracle for and an oracle to factorise integers of the form for , it constructs a composition tree for G.
As we shall explain in Section 2.3, the oracle to factorise integers of the form 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 as composition factors but, since no short presentations are known for the groups , 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 that may be rewritten over an extension field for some . 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 for certain 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 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 . 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 of elements of G and returns an encoding of gh; a
Deciding simplicity
Theorem 3.1 There is a Monte Carlo polynomial-time algorithm that takes as input a nonabelian black-box group and and outputs one of the following: , the name of G, and a constructive membership algorithm for G; and ; 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 – and . We extend (or abuse) this notation by applying it to arbitrary subgroups of maximal subgroups in classes – . For example, we view every reducible matrix group as a member of class . Viewed in this way, the main result of [1] is that every subgroup G of either lies in at least one of – , or it lies in or , 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 and outputs a composition tree for G.
- (1)
Do one of the following:
- (i)
construct an effective epimorphism , for some group ; or
- (ii)
prove that G is simple, in which case G becomes a leaf in the tree.
- (i)
- (2)
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