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# Topic: Group theory

###### In the News (Sat 15 Jun 19)

 20: Group Theory and Generalizations Group theory can be considered the study of symmetry: the collection of symmetries of some object preserving some of its structure forms a group; in some sense all groups arise this way. Groups acting on topological spaces are the basis of equivariant topology and homotopy theory in Algebraic Topology. Nielsen's theorem: subgroups of free groups are free. www.math.niu.edu /~rusin/known-math/index/20-XX.html   (2774 words)

 Group theory - Wikipedia, the free encyclopedia Group theory is that branch of mathematics concerned with the study of groups. A common foundation for the theory of equations on the basis of the group of permutations was found by mathematician Lagrange (1770, 1771), and on this was built the theory of substitutions. Group theory is also very important to the field of chemistry, where it is used to assign symmetries to molecules. en.wikipedia.org /wiki/Group_theory   (1367 words)

 PlanetMath: group Groups often arise as the symmetry groups of other mathematical objects; the study of such situations uses group actions. See Also: subgroup, cyclic group, simple group, symmetric group, free group, ring, field, group homomorphism, Lagrange's theorem, identity element, proper subgroup, groupoid, fundamental group, topological group (obsolete), Lie group, Proof: The orbit of any element of a group is a subgroup, locally cyclic group, existence of Hilbert class field, abelian group, This is version 17 of group, born on 2001-08-29, modified 2006-03-14. planetmath.org /encyclopedia/Group.html   (314 words)

 Group theory Summary Group theory is one of a number of branches of mathematics that have proven useful to chemists and physicists in their work. Group theory is that branch of mathematics concerned with the study of groups. A common foundation for the theory of equations on the basis of the group of permutations was found by mathematician Lagrange (1770, 1771), and on this was built the theory of substitutions. www.bookrags.com /Group_theory   (3252 words)

 Group theory Möbius in 1827, although he was completely unaware of the group concept, began to classify geometries using the fact that a particular geometry studies properties invariant under a particular group. Hölder was to prove it in the context of abstract groups in 1889. At that time the only known groups were groups of permutations and even this was a radically new area, yet Cayley defines an abstract group and gives a table to display the group multiplication. www-groups.dcs.st-and.ac.uk /~history/HistTopics/Development_group_theory.html   (1464 words)

 An Introduction to GROUP THEORY The theory does not concern itself with what a and b actually are nor with what the operation symbolized by • actually is. By taking this abstract approach group theory deals with many mathematical systems at once. Group theory is a clear example of abstraction in modern mathematics. This is because in group theory commutativity is not assumed. members.tripod.com /~dogschool/groups.html   (1440 words)

 !GROUP THEORY! Along with the set of the group, thereis a binary operation such that when one element operates on another thenthe result is still an element in the set. Given a finite group (G,*)-where G isthe set of the group and the * is the binary operation; and give a andb being elements of the set G; then a*b=b*a. Whena free group is given a number of operations then the most general groupthat can be built from them using the powers of these elements and theinverses of the powers. www.geocities.com /CapeCanaveral/Hangar/9302/group.html   (3637 words)

 Ring Theory In contrast to commutative ring theory, which as we have seen grew from number theory, non-commutative ring theory developed from an idea which, at the time of its discovery, was heralded as a great advance in applied mathematics. The greatest early contributor to the theory of non-commutative rings was the Scottish mathematician Wedderburn. The Wedderburn theory was extended to non-commutative rings satisfying both ascending and descending finiteness conditions (called chain conditions) by Artin in 1927. www-groups.dcs.st-and.ac.uk /~history/HistTopics/Ring_theory.html   (1890 words)

 Group Theory (on ScienceFizz.com)   (Site not responding. Last check: 2007-10-30) Group theory section of the mathematics e-print arXiv. 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. Group Theory is a branch of algebra, but has strong connections with almost all parts of mathematics. www.sciencefizz.com /Math/Algebra/Group_Theory   (485 words)

 Group Theory and Physics Physics uses that part of Group Theory known as the theory of representations, in which matrices acting on the members of a vector space is the central theme. In introducing a subject, especially one as abstract as Group Theory, it is important to begin with concrete, explicit examples and not with general principles. This group is too small to show some of the more interesting aspects of groupp structure, but it is indeed a group, and an important one. www.du.edu /~jcalvert/phys/groups.htm   (5735 words)

 cmm   (Site not responding. Last check: 2007-10-30) Muted Group Theory is a critical theory because it is concerned with power and how it is used against people. Muted Group Theory begins with the premise that language is culture bound, and because men have more power than women, men have more influence over the language, resulting in language with a male-bias. Instead, the theory claims that men risk losing their dominant position if they listen to women, incorporate their experiences in the language, and allow women to be equal partners in language use and creation. oregonstate.edu /instruct/theory/mutedgrp.html   (356 words)

 Group theory - Conservapedia Group theory is the study of mathematical groups, including their symmetries and permutations. A second source was the theory of algebraic equations, leading to the study of permutations, also beginning in the late 1700s. It is generally recognised that Group Theory began with Evariste Galois in the early 19th century, who recognised some patterns in the roots of Quintics. www.conservapedia.com /Group_theory   (241 words)

 Group Theory Group Theory is one of the most powerful mathematical tools used in Quantum Chemistry and Spectroscopy. The key to applying Group Theory is to be able to identify the "Point Group" of the molecule i.e. Therefore the first step in applying Group Theory to molecular properties is to identify the complete set of Symmetry Elements possessed by the molecule. www.science.siu.edu /chemistry/tyrrell/group_theory/sym1.html   (487 words)

 Seminar "Group theory and topology" Finitely generated groups G become geometric objects when endowed with the word metric (depending on the generating set): the distance between a and b from G is the length of the shortest word representing a^{-1}b. The existence of a bounded-simple 2-generated group, containing a free non-cyclic subgroup, and the existence of an infinite simple bounded-generated 2-generated group are proven. group G and every homomorphism from F to a free group of rank 2 extends to G. Then F is a retract of G. math.vanderbilt.edu /~msapir/altopfall02.html   (942 words)

 The Math Forum - Math Library - Group Theory Group Theory is a branch of algebra, but has strong connections with almost all parts of mathematics. Group theory takes an abstract approach, dealing with many mathematical systems at once and requiring only that a mathematical system obey a few simple rules, seeking then to find properties common to all systems that obey these few rules. A short article designed to provide an introduction to finite groups - their internal properties: all those results about group theory for which a consideration of the order of elements is a central part of the question. mathforum.org /library/topics/group_theory   (2240 words)

 Higher dimensional group theory This is a diagram of the historical develpment of notions of higher dimensional algebra, from the background to group theory to Pursuing Stacks. In view of the importance of group theory in mathematics, a higher dimensional theory should be of great significance, and it would be foolish to turn away from a chance to investigate it. In the work of Brown and Loday, the new determination of the third homotopy group of the suspension of a K(G,1) as the kernel of a 'commutator morphism' from the non abelian tensor square of G, is in fact exercise 1 in the new theory. www.bangor.ac.uk /~mas010/hdaweb2.htm   (5854 words)

 Chapt. One, IV.4. Group Theory Music theory is a classification of families of notes and their arrangements in certain patterns. A familiar group is the group of integers: -1, 0, 1, 2, 3, etc. An Operation for this group is addition: 2 + 3 = 5. It seems to be a mathematical impossibility that this many agreements of his constructs with group theory just happened by accident, and is virtual proof that he was somehow playing around with these concepts. members.aol.com /chang8828/grouptheory.htm   (4450 words)

 Category:Group theory - Wikipedia, the free encyclopedia Articles and media on this topic in other Wikimedia projects can be found at: Commons Category Group theory In mathematics, a group is a set, together with a binary operation satisfying certain axioms, detailed in the group article. Fixed points of isometry groups in Euclidean space en.wikipedia.org /wiki/Category:Group_theory   (114 words)

 Group Theory The Group Theory lecture course will be given as 1 lecture per week (4:15-5:45 pm) and will be given in english. Joshi, Elements of Group theory for Physicists, Wiley Eastern New Delhi, 1982. Hall, Group theory and Symmetry in chemistry, McGraw-Hill, New York, 1969. wwwitp.physik.tu-berlin.de /lehress01/grouptheory   (98 words)

 Open Directory - Science: Math: Algebra: Group Theory 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. Group Action Forum - Association for the study of the theory of transformation groups and related topics. Group Pub Forum Home Page - These are the community pages for Group Theory, the mathematics of symmetry. dmoz.org /Science/Math/Algebra/Group_Theory   (626 words)

 Group Theory   (Site not responding. Last check: 2007-10-30) The most familiar group in physics is the rotation group governing rotations in three-dimensional space. The theory of spinors is developed in Chapter IV; the group SL(2,C) is studied in detail. Later, in the last two chapters of the book, some applications of this oscillator formalism to hadronic phenomenology are given, for example, to give an explanation of the mass spectra of hadrons and the study the deformation properties of relativistic hadrons. www2.physics.umd.edu /~yskim/home/groth.html   (609 words)

 Theory and Semantics Group   (Site not responding. Last check: 2007-10-30) The work of the Theory and Semantics Group is centred around mathematical models of a variety of languages and logics. These include finite model theory and its connection to the study of computational complexity; the theory of databases; the complexity of games and the expressive power of logical formalisms. The syntax and semantics of the lambda calculus, general recursion theory, sheaf models for intuitionistic theories, general categorical logic and topos theory, the effective topos, other realizability toposes and constructive mathematics as developed in such frameworks, topos theoretic models for polymorphism, linear logic and game theoretic semantics. www.cl.cam.ac.uk /Research/TSG   (1103 words)

 Group Theory & Rubik's Cube Group theory is the study of the algebra of transformations and symmetry. Given an element x of a group G, the orbit of x is the set of all elements of G which are generated by x, i.e. A representation of a group G is a set of matrices M which are homomorphic to the group. akbar.marlboro.edu /~mahoney/courses/Spr00/rubik.html   (3602 words)

 Lesson: Nat. of Sci. mini-lesson: Theory, Theory Also, because a hypothesis may, after some indefinite degree of confirmation, rise to the level of "theory" (or part of a theory), references to a hypothesis as a "theory" should be presented with qualifying quotation marks. In the second situation, after class discussion of their evaluations, ask students to indicate which of the 5 scenarios is technically a scientific theory, and which are not, and the reason(s) for saying that (unless they have already pointed it out during discussion in the activity). And, because "Theory E" is a direct product of an established religion, it is actually illegal to include it as a scientific alternative in any public school science class in the USA, as concluded by the US Supreme Court. www.indiana.edu /~ensiweb/lessons/theory.html   (1394 words)

 Muted Group Theory by Cheris Kramarae Muted Group theory attempts to explain why certain groups in society are muted which means they are either silent or not heard. From my perspective, Muted Group theory explains why I, as a woman, have had a difficult time being accepted into and becoming a part of an organization whose rules have been constructed primarily by a white, male-dominated world. Another limitation as I see it is although this theory explains who, what and where most muted groups can be found, it appears that even the very powerful men who muted group theory refer to can also find themselves in a powerless situations at times. www.colorado.edu /communication/meta-discourses/Papers/App_Papers/Baer.htm   (2769 words)

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