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Algorithms in algebraic number theory
 Bull. Amer. Math. Soc
, 1992
"... Abstract. In this paper we discuss the basic problems of algorithmic algebraic number theory. The emphasis is on aspects that are of interest from a purely mathematical point of view, and practical issues are largely disregarded. We describe what has been done and, more importantly, what remains to ..."
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Cited by 40 (3 self)
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Abstract. In this paper we discuss the basic problems of algorithmic algebraic number theory. The emphasis is on aspects that are of interest from a purely mathematical point of view, and practical issues are largely disregarded. We describe what has been done and, more importantly, what remains to be done in the area. We hope to show that the study of algorithms not only increases our understanding of algebraic number fields but also stimulates our curiosity about them. The discussion is concentrated of three topics: the determination of Galois groups, the determination of the ring of integers of an algebraic number field, and the computation of the group of units and the class group of that ring of integers. 1.
Generalizing boolean satisfiability II: Theory
, 2004
"... This is the second of three planned papers describing zap, a satisfiability engine that substantially generalizes existing tools while retaining the performance characteristics of modern high performance solvers. The fundamental idea underlying zap is that many problems passed to such engines contai ..."
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Cited by 11 (2 self)
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This is the second of three planned papers describing zap, a satisfiability engine that substantially generalizes existing tools while retaining the performance characteristics of modern high performance solvers. The fundamental idea underlying zap is that many problems passed to such engines contain rich internal structure that is obscured by the Boolean representation used; our goal is to define a representation in which this structure is apparent and can easily be exploited to improve computational performance. This paper presents the theoretical basis for the ideas underlying zap, arguing that existing ideas in this area exploit a single, recurring structure in that multiple database axioms can be obtained by operating on a single axiom using a subgroup of the group of permutations on the literals in the problem. We argue that the group structure precisely captures the general structure at which earlier approaches hinted, and give numerous examples of its use. We go on to extend the DavisPutnamLogemannLoveland inference procedure to this broader setting, and show that earlier computational improvements are either subsumed or left intact by the new method. The third paper in this series discusses zap’s implementation and presents experimental performance results. 1.
Some topics in asymptotic group theory
 Groups, Combinatorics and Geometry, volume 165 of LMS Lecture Notes
, 1992
"... There is nothing unusual about asymptotics in fmite group theory: there are a number of known (or even wellknown) asymptotic results. While these are not really the subject of this paper, it seems appropriate to begin with some especially intriguing examples (the fIrst and last of which will be nee ..."
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Cited by 7 (4 self)
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There is nothing unusual about asymptotics in fmite group theory: there are a number of known (or even wellknown) asymptotic results. While these are not really the subject of this paper, it seems appropriate to begin with some especially intriguing examples (the fIrst and last of which will be need later). 1.1. If P is prime then the number of isomorphism classes of groups of order pk.l. k3 _ 6k.l.f3 + 0(k8/3) is at least pZ7 (Higman [HiD, and asymptotically::: p2 (Sims [SiD.
Boolean Routing on Cayley Networks
, 1997
"... We study Boolean routing on Cayley networks. Let K(MG ) denote the Kolmogorov complexity of the multiplication table of a group G of order n. We show that O(maxfd log n; K(MG )g) memory bits per local router (hence a total of O(nmaxfd log n; K(MG )g) memory bits) are sufficient to do Boolean routin ..."
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Cited by 6 (3 self)
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We study Boolean routing on Cayley networks. Let K(MG ) denote the Kolmogorov complexity of the multiplication table of a group G of order n. We show that O(maxfd log n; K(MG )g) memory bits per local router (hence a total of O(nmaxfd log n; K(MG )g) memory bits) are sufficient to do Boolean routing on the Cayley network constructed from the group G and a set of d generators. It is a consequence of the classification theorem for finite simple groups that K(MG ) 2 O(log 3 n). Even better complexity bounds are known for classes of groups. E.g., if n is the power of a prime p then K(MG ) 2 O(log 3 n= log 2 p); if n is squarefree or G is abelian then K(MG ) 2 O(log n). A lower bound \Omega\Gamma nd log d + n log n) on the total number of memory bits is given in [4]. 1980 Mathematics Subject Classification: 68Q99 CR Categories: C.2.1 Key Words and Phrases: Boolean routing, Cayley Network, Kolmogorov complexity, Multiplication Table. Carleton University, School of Computer Scie...
Generators for finite groups with a unique minimal normal subgroup
 MR98m:20034, Zbl 0895.20027
, 1997
"... Assume that a finite group G has a unique minimal normal subgroup, say N. We prove that if the order of N is large enough then the following is true: If d randomly chosen elements generate G modulo N, then these elements almost certainly generate G itself. 1. Introduction. For any finite group G, le ..."
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Cited by 1 (1 self)
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Assume that a finite group G has a unique minimal normal subgroup, say N. We prove that if the order of N is large enough then the following is true: If d randomly chosen elements generate G modulo N, then these elements almost certainly generate G itself. 1. Introduction. For any finite group G, let d(G) be the smallest cardinality of a generating set of G and let φG(d) denote the number of dbasis, that is, ordered dtuples (g1,...,gd) of elements of G that generate G. The number PG(d) = φG(d) G  d
Some Measures of Finite Groups Related to Permutation Bases
"... I define three "measures" of the complicatedness of a finite group in terms of bases in permutation representations of the group, and consider their relationships to other measures. ..."
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I define three "measures" of the complicatedness of a finite group in terms of bases in permutation representations of the group, and consider their relationships to other measures.
Groups of a SquareFree Order
, 2010
"... Hölder’s formula for the number of groups of a squarefree order is an early advance in the enumeration of finite groups. This paper gives a structural proof of Hölder’s result that is accessible to undergraduates. We introduce a number of group theoretic concepts such as nilpotency, the Fitting sub ..."
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Hölder’s formula for the number of groups of a squarefree order is an early advance in the enumeration of finite groups. This paper gives a structural proof of Hölder’s result that is accessible to undergraduates. We introduce a number of group theoretic concepts such as nilpotency, the Fitting subgroup, and extensions. These topics, which are usually not covered in undergraduate group theory, feature in the proof of Hölder’s result and have wide applicability in group theory. Finally, we remark on further results and conjectures in the enumeration of finite groups.
Lectures on derangements
, 2011
"... These are notes from my lectures at the Pretty Structures conference at the Institut Henri Poincaré in Paris, in early May 2011. I had planned to give three talks about derangements; but in the event, the third lecture was devoted to the topic of synchronization, and as a result some of this materia ..."
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These are notes from my lectures at the Pretty Structures conference at the Institut Henri Poincaré in Paris, in early May 2011. I had planned to give three talks about derangements; but in the event, the third lecture was devoted to the topic of synchronization, and as a result some of this material was not covered in the talks. A general reference on permutation groups is my book [2]. Notes on synchronization (containing far more than I put into the single lecture in Paris) can be found at