Gourt > Science > Math > Algebra > Group Theory (1)
Group theory is that branch of mathematics concerned with the study of groups. It has several applications in physics and chemistry. Please refer to the glossary of group theory for the definitions of terms used throughout group theory. See also the list of group theory topics.
Groups are used throughout mathematics, often to capture the internal symmetry of other structures, in the form of automorphism groups. An internal symmetry of a structure is usually associated with an invariant property; the set of transformations that preserve this invariant property, together with the operation of composition of transformations, form a group called a symmetry group.
In Galois theory, which is the historical origin of the group concept, one uses groups to describe the symmetries of the equations satisfied by the solutions to a polynomial equation. The solvable groups are so-named because of their prominent role in this theory.
More on [ Group theory ]
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Abstract groups - The abstract group concept. Material based on lectures by Peter Neumann.
ArXiv Front: GR Group Theory - Group theory section of the mathematics e-print arXiv.
Binary Coordinate Systems - Article by Steven H. Cullinane. Defines Cartesian coordinate systems for small finite geometries over the two-element field, and discusses geometrically simple generators for affine group actions.
Broue's Abelian Defect Group Conjecture - A record of which groups and blocks have been proved to satisfy the conjecture, maintained by Jeremy Rickard.
Computational Tools for Group Theory - Describes work to create a program that could be used to generate, identify, and analyze finite groups presented in the form of a Cayley Table as well as visualize the groups that are generated.
Meta Description: [ Thesis for Jeffrey Barr, SDSU, Spring 2005 ]
Coxeter/Weyl Tutorial - A Tutorial Introduction to the Coxeter and Weyl Packages, a pair of Maple packages for working with root systems, finite Coxeter groups and Weyl characters
GAP - Groups, Algorithms and Programming - GAP is a free system for computational discrete algebra.
Meta Description: [ GAP system for computational discrete algebra, especially computational group theory ]
Group Action Forum - Association for the study of the theory of transformation groups and related topics. Members, news, events, publications.
Meta Description: [ GAF - Group Action Forum - an international mathematical association ]
Group Pub Forum Home Page - These are the community pages for Group Theory, the mathematics of symmetry. Group Theory is a branch of algebra, but has strong connections with almost all parts of mathematics.
Meta Description: [ These are the community pages for
Group Theory, the mathematics of symmetry. Group Theory is a branch
of algebra, but has strong connections with almost all
parts of mathematics.
]
Group Theory - On-line book by Prederic Cvitanovic. Available both in 'psd' and 'ps' formats.
International Society for Group Theory in Cognitive Science - Group theory in: Robotics, Problem-Solving, Planning, Learning, Language, Perception, Art, Design, Engineering, Manufacturing, Epistemology, Measurement, Computation, Neuroscience, Anthropology, Semiotics.
Introduction to Group Theory - A fairly easy to understand tutorial. Fourteen sections, including groups, Cayley tables, subgroups, cosets, Lagrange's theorem, cyclic groups and subgroups, permutations, and Rubik's cube.
Mod-2 Cohomology of 2-Groups - Computer calculation of the mod-2 cohomology of groups of order 8, 16, 32 and 64.
New York Group Theory Cooperative - Software (Magnus), preprints, meetings, links. Magnus - a graphically-oriented system for computational group theory - allows one to explore and experiment with abstract groups without the need for learning yet another programming language. Magnus is freely available software (GPL) based on Tcl/Tk.
Open Problems in Group Theory - Part of the Magnus project. Contains over 150 problems in group theory, both well known and relatively new.
Papers by R. E. Borcherds - Including proof of the Moonshine Conjecture (TeX, DVI, PDF).
PARGAP - Parallel GAP/MPI (ParGAP/MPI), a share package for GAP. UNIX (or Cygwin/Windows). Download source and documentation by FTP.
Permutation Group Problems - Compiled by Peter Cameron.
Meta Description: [ Research problems on permutation groups,
with commentary ]
Permutation Groups Resources - Web-based resources for permutation groups and related areas in group theory and combinatorics.
Meta Description: [ Web-based resources for permutation groups
and related areas ]
404
Semigroup Theory - Directory of home pages and conferences maintained at Southampton University.
Symmetric Presentations - Some presentations of groups (symmetric or otherwise).
Symmetry - The main point group symmetries of interest to defect physics by operation (reflection, rotations etc) and classification (trigonal, and cubic). Most point groups also have the associated character table on-line.
Symmetry Groups in Maple V - A worksheet by W.D. Joyner.
The Lie algebras su(N), an introduction - Publication: An Introduction to the Lie Algebras su(N). By Walter Pfeifer, Switzerland. A free copy can be ordered.
Meta Description: [ An Introduction to the Lie Algebras su(N). It is
conceived to give directly a concrete idea of this algeba with numerous numerical tests.
This publication can be ordered for free. ]
The Moonshine Page - Bibliography, meetings.
404
The North British Quantum Groups Collective - The North British Quantum Groups Collective meets regularly to discuss research in quantum groups, with the aim of disseminating recent results and exchanging ideas and information on the subject.
The World of Groups - Part of the World Wide Algebra project. Open problems in combinatorial group theory, a list of personal web pages, conferences and seminars, and useful links.
Topological Methods in Group Theory - Table of contents only; draft chapters can be downloaded by arrangement.
Trinomials with Interesting Galois Groups - Parametrisation of trinomials with Galois groups contained in the simple group G168.
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