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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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Cited by 13 (2 self)
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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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Cited by 2 (0 self)
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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...
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
The Classification of the Finite Simple Groups: An Overview
 MONOGRAFÍAS DE LA REAL ACADEMIA DE CIENCIAS DE ZARAGOZA. 26: 89–104, (2004)
, 2004
"... ..."
unknown title
"... 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 t ..."
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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 ..."
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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
MORE ON THE SYLOW THEOREMS
"... Several alternative proofs of the Sylow theorems are collected here. Section 2 has a proof ..."
Abstract
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Several alternative proofs of the Sylow theorems are collected here. Section 2 has a proof
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
 Add to MetaCart
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