Colin Ramsay

BREADTH-FIRST SEARCH AND THE ANDREWS–CURTIS CONJECTURE

Andrews and Curtis conjectured in 1965 that every balanced presentation of the trivial group can be transformed into a standard presentation by a finite sequence of elementary transformations. Recent computational work by Miasnikov and Myasnikov on this problem has been based on genetic algorithms. We show that a computational attack based on a breadth-first search of the tree of equivalent presentations is also viable, and seems to outperform that based on genetic algorithms. It allows us to extract shorter proofs (in some cases, provably shortest) and to consider the length thirteen case for two generators. We prove that, up to equivalence, there is a unique minimum potential counterexample.

Nice Efficient Presentions for all Small Simple Groups and their Covers

Prior to this paper, all small simple groups were known to be efficient, but the status of four of their covering groups was unknown. Nice, efficient presentations are provided in this paper for all of these groups, resolving the previously unknown cases. The authors’ presentations are better than those that were previously available, in terms of both length and computational properties. In many cases, these presentations have minimal possible length. The results presented here are based on major amounts of computation. Substantial use is made of systems for computational group theory and, in particular, of computer implementations of coset enumeration. To assist in reducing the number of relators, theorems are provided to enable the amalgamation of power relations in certain presentations. The paper concludes with a selection of unsolved problems about efficient presentations for simple groups and their covers.

On the spectrum of Steiner (v,k,t) trades (II)

Abstract. A (v,k,t) trade T=T1−T2 of volume m consists of two disjoint collections T1 and T2 each containing m blocks (k-subsets) such that each t-subset is contained in the same number of blocks in T1 and T2. If each t-subset occurs at most once in T1, then T is called a Steiner (k,t) trade. In this paper, we continue our investigation of the spectrum S(k,2); that is, the set of allowable volumes of Steiner (k,t) trades when t=2. By using the concept of linked trades, we show that 2k+1∈S(k,2) precisely when k∈{3, 4, 7}.

Completely separating systems of k -sets

Dickson (1969) introduced the notion of a completely separating set system. We study such systems with the additional constraint that each set in the system has the same size. Let T denote an n-set. We say that a subset S of T separates i from j if i ∈ S and j ∉ S. A collection of k-sets script c sign is called a (n, k)-separator if, for each ordered pair (i,j) ∈ T × T with i ≠ j, there is a set S ∈ script c sign which separates i from j. Let R(n, k) denote the size of a smallest (n, k)-separator. For n ≥ k(k - 1) we show that R(n, k) = [2n/k]. We also show that R(2,2)≤2m and demonstrate various recursive relationships that are used to determine the exact values of R(n, k) for k ≤ 5.

On the Efficiency of the Simple Groups of Order Less Than a Million and Their Covers

There is much interest in finding short presentations for the finite simple groups. In a previous paper we produced nice efficient presentations for all small simple groups and for their covering groups. Here we extend those results from simple groups of order less than 100,000 up to order one million, but we leave one simple group and one covering group for which the efficiency question remains unresolved. We give presentations that are better than what was previously available, in terms of length and in terms of computational properties, in the process answering two previously unresolved problems about the efficiency of covering groups of simple groups. Our results are based on major amounts of computation. We make substantial use of systems for computational group theory and, in particular, of computer implementations of coset enumeration.

On the spectrum of [v, k, t] trades

Abstract A [v, k, t] trade of volume m consists of two disjoint collections T1 and T2, each of m k-subsets of a v-set V, such that each t-subset of V is contained in the same number of blocks of T1 and T2, and each element of V is contained in at least one block of T1. We study [v, k, t] trades, and investigate their spectrum (i.e., the collections of allowable volumes), using both theoretical techniques and computer-based searches.

Groups St Andrews 2005: On proofs in finitely presented groups

Given a finite presentation of a group G, proving properties of G can be difficult. Indeed, many questions about finitely presented groups are unsolvable in general. Algorithms exist for answering some questions while for other questions algorithms exist for verifying the truth of positive answers. An important tool in this regard is the Todd-Coxeter coset enumeration procedure. It is possible to extract formal proofs from the internal working of coset enumerations. We give examples of how this works, and show how the proofs produced can be mechanically verified and how they can be converted to alternative forms. We discuss these automatically produced proofs in terms of their size and the insights they offer. We compare them to hand proofs and to the simplest possible proofs. We point out that this technique has been used to help solve a longstanding conjecture about an infinite class of finitely presented groups.

Groups St Andrews 2001 in Oxford: Short balanced presentations of perfect groups

We report some initial results from an investigation of short balanced presentations of perfect groups. We determine the minimal length 2-generator balanced presentations for SL2(5) and SL2(7) and we show that M̂22, the full covering group of the sporadic simple group M22, has deficiency zero. We give presentations for SL2(7) and M̂22 that are both new and of minimal length. We also determine the shortest 2-generator presentations for an infinite perfect group. This is done in the context of a complete classification of short 2-generator balanced presentations of perfect groups in terms of canonic presentations.

Andrews-Curtis and Todd-Coxeter proof words

Andrews and Curtis have conjectured that every balanced presentation of the trivial group can be transformed into a standard presentation by a finite sequence of elementary transformations. It can be difficult to determine whether or not the conjecture holds for a particular presentation. We show that the utility PEACE, which produces proofs based on Todd-Coxeter coset enumeration, can produce Andrews-Curtis proofs.

Parallel COSET enumeration using threads

Coset enumeration is one of the basic tools for investigating finitely presented groups . Many enumerations require significant resources , in terms of CPU time or memory space . We develop a fully functional parallel coset enumeration procedure and we discuss some of the issues involved in such parallelisation using the POSIX threads library. Our results can equally well be applied to any master-slave parallel activity exhibiting a medium level of granularity.