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13
Modular Reasoning in Isabelle
, 1999
"... The concept of locales for Isabelle enables local definition and assumption for interactive mechanical proofs. Furthermore, dependent types are constructed in Isabelle/HOL for first class representation of structure. These two concepts are introduced briefly. Although each of them has proved use ..."
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The concept of locales for Isabelle enables local definition and assumption for interactive mechanical proofs. Furthermore, dependent types are constructed in Isabelle/HOL for first class representation of structure. These two concepts are introduced briefly. Although each of them has proved useful in itself, their real power lies in combination. This paper illustrates by examples from abstract algebra how this combination works and argues that it enables modular reasoning.
Group Theory
, 2003
"... The first version of these notes was written for a firstyear graduate algebra course. As in most such courses, the notes concentrated on abstract groups and, in particular, on finite groups. However, it is not as abstract groups that most mathematicians encounter groups, but rather as algebraic gro ..."
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Cited by 6 (0 self)
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The first version of these notes was written for a firstyear graduate algebra course. As in most such courses, the notes concentrated on abstract groups and, in particular, on finite groups. However, it is not as abstract groups that most mathematicians encounter groups, but rather as algebraic groups, topological groups, or Lie groups, and it is not just the groups themselves that are of interest, but also their linear representations. It is my intention (one day) to expand the notes to take account of this, and to produce a volume that, while still modest in size (c200 pages), will provide a more comprehensive introduction to group theory for beginning graduate students in mathematics, physics, and related fields. Please send comments and corrections to me at math0 at jmilne.org. v2.01 (August 21, 1996). First version on the web; 57 pages. v2.11 (August 29,2003). Fixed many minor errors; numbering unchanged; 85 pages.
Cross domain mathematical concept formation
 In Proceedings of the AISB00 Symposium on Creative & Cultural Aspects and Applications of AI & Cognitive Science
, 2000
"... Many interesting concepts in mathematics are essentially ‘crossdomain ’ in nature, relating objects from more than one area of mathematics, e.g. prime order groups. These concepts are often vital to the formation of a mathematical theory. Often, the introduction of crossdomain concepts to an inves ..."
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Cited by 5 (1 self)
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Many interesting concepts in mathematics are essentially ‘crossdomain ’ in nature, relating objects from more than one area of mathematics, e.g. prime order groups. These concepts are often vital to the formation of a mathematical theory. Often, the introduction of crossdomain concepts to an investigation seems to exercise a mathematician’s creative ability. The HR program, (Colton et al., 1999), proposes new concepts in mathematics. Its original implementation was limited to working in one mathematical domain at a time, so it was unable to create crossdomain concepts. Here, we describe an extension of HR to multiple domains. Crossdomain concept formation is facilitated by generalisation of the data structures and heuristic measures employed by the program, and the implementation of a new production rule. Results achieved include generation of the concepts of prime order groups, graph nodes of maximal degree and an interesting class of graph. 1
Applications Of Burnside Rings In Elementary Group Theory
"... : This is a report on some of the results which appear in [DSY 90]. A canonical ring homomorphism from the Burnside ring\Omega\Gamma C) of a finite cyclic group C into the Burnside ring\Omega\Gamma G) of any finite group G of the same order is exhibited and it is shown that many results from eleme ..."
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: This is a report on some of the results which appear in [DSY 90]. A canonical ring homomorphism from the Burnside ring\Omega\Gamma C) of a finite cyclic group C into the Burnside ring\Omega\Gamma G) of any finite group G of the same order is exhibited and it is shown that many results from elementary finite group theory, in particular those claiming certain congruence relations, are simple consequences of the existence of this map. Theorem: Let G be a finite group and let C denote the cyclic group of the same order n. There exists a ring homomorphism ff = ff(G) : \Omega\Gamma C) \Gamma! \Omega\Gamma G) from the Burnside ring\Omega\Gamma C) of the cyclic group C into the Burnside ring \Omega\Gamma G) of the group G with the following property: ffl for every subgroup U G of G and every element x 2 \Omega\Gamma C) one has 'U (ff(x)) = 'C jU j (x) where 'U (ff(x)) denotes the number of U invariant elements in the virtual Gset ff(x) and C jU j denotes the unique subgroup of...
Small Base Groups, Large Base Groups and the Case of Giants
"... Abstract. Let G be a finite permutation group acting on a set Γ. A base for G is a finite sequence of elements of Γ whose pointwise stabiliser in G is trivial. Most (families of) finite permutation groups admit a base that grows very slowly as the degree of the group increases. Such groups are known ..."
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Abstract. Let G be a finite permutation group acting on a set Γ. A base for G is a finite sequence of elements of Γ whose pointwise stabiliser in G is trivial. Most (families of) finite permutation groups admit a base that grows very slowly as the degree of the group increases. Such groups are known as small base and very efficient algorithms exist for dealing with them. However, some families of permutation groups, such as the symmetric groups, do not admit a small base. Dealing with these socalled large base groups is a fascinating area of current research. This thesis explores two closely interrelated strands of modern group theory. Initially, the focus is on identifying the large base primitive permutation groups, which can be achieved by making use of two landmark results in finite group theory: The Classification of Finite Simple Groups and the O’NanScott Theorem for primitive permutation groups. Focus then shifts to algorithmic aspects of large base groups, in particular to the family known as the giants. We cover details such as recognition of large base Galois groups, generation of random elements of finite groups and give details of the very new paradigm of algorithms for black box groups. We conclude with an investigation into the constructive recognition problem for large base black box groups.
unknown title
"... Version 3.10 September 9, 2010The first version of these notes was written for a firstyear graduate algebra course. As in most such courses, the notes concentrated on abstract groups and, in particular, on finite groups. However, it is not as abstract groups that most mathematicians encounter group ..."
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Version 3.10 September 9, 2010The first version of these notes was written for a firstyear graduate algebra course. As in most such courses, the notes concentrated on abstract groups and, in particular, on finite groups. However, it is not as abstract groups that most mathematicians encounter groups, but rather as algebraic groups, topological groups, or Lie groups, and it is not just the groups themselves that are of interest, but also their linear representations. It is my intention (one day) to expand the notes to take account of this, and to produce a volume that, while still modest in size (c200 pages), will provide a more comprehensive introduction to group theory for beginning graduate students in mathematics, physics, and related fields. BibTeX information
unknown title
"... Version 3.10 September 24, 2010The first version of these notes was written for a firstyear graduate algebra course. As in most such courses, the notes concentrated on abstract groups and, in particular, on finite groups. However, it is not as abstract groups that most mathematicians encounter grou ..."
Abstract
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Version 3.10 September 24, 2010The first version of these notes was written for a firstyear graduate algebra course. As in most such courses, the notes concentrated on abstract groups and, in particular, on finite groups. However, it is not as abstract groups that most mathematicians encounter groups, but rather as algebraic groups, topological groups, or Lie groups, and it is not just the groups themselves that are of interest, but also their linear representations. It is my intention (one day) to expand the notes to take account of this, and to produce a volume that, while still modest in size (c200 pages), will provide a more comprehensive introduction to group theory for beginning graduate students in mathematics, physics, and related fields. BibTeX information
The Classification of the Finite Simple Groups: An Overview
 MONOGRAFÍAS DE LA REAL ACADEMIA DE CIENCIAS DE ZARAGOZA. 26: 89–104, (2004)
, 2004
"... ..."
MORE ON THE SYLOW THEOREMS
"... Several alternative proofs of the Sylow theorems are collected here. Section 2 has a proof ..."
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Several alternative proofs of the Sylow theorems are collected here. Section 2 has a proof
A METHOD FOR DETERMINING THE MOD2k BEHAVIOUR OF RECURSIVE SEQUENCES, WITH APPLICATIONS TO SUBGROUP COUNTING
"... Abstract. We present a method to obtain congruences modulo powers of 2 for sequences given by recurrences of finite depth with polynomial coefficients. We apply this method to Catalan numbers, Fuß–Catalan numbers, and to subgroup counting functions associated with Hecke groups and their lifts. This ..."
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Abstract. We present a method to obtain congruences modulo powers of 2 for sequences given by recurrences of finite depth with polynomial coefficients. We apply this method to Catalan numbers, Fuß–Catalan numbers, and to subgroup counting functions associated with Hecke groups and their lifts. This leads to numerous new results, including many extensions of known results to higher powers of 2.