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Topic: Elementary group theory

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	Spartanburg SC \| GoUpstate.com \| Spartanburg Herald-Journal (Site not responding. Last check: 2007-11-05)
		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.
		In Philosophy, Ernst Cassirer related the theory of group to the theory of perception as described by Gestalt Psychology; Perceptual Constancy is taken to be analogous to the invariants of group theory.
	www.goupstate.com /apps/pbcs.dll/section?category=NEWS&template=wiki&text=group_theory (1842 words)

	Kids.Net.Au - Encyclopedia > Elementary group theory
		The order of a group (G,) is the number of elements in G (for a finite group), or the cardinality of the group* if G is not finite.
		A proper subgroup of a group G is a subgroup which is not identical to G.
		Define the index of a subgroup H of a group G (written "[G:H]") to be the number of distinct left cosets of H in G.
	www.kids.net.au /encyclopedia-wiki/el/Elementary_group_theory (1576 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.
		The methods of group theory should, therefore, be applicable in the study of the behavior of symmetrical molecules such as AB The application of group theory to chemical molecules is called chemical group theory.
		The theory of groups is a branch of mathematics in which one does something to something and then compares the results with the result of doing the same thing to something else, or something else to the same thing.
	www.bookrags.com /Group_theory (3252 words)

	Group Theory at the Library of Math (Free Online Mathematics) (Site not responding. Last check: 2007-11-05)
		Basically a group is a set together with a single operation that satisfies certain properties: (1) there must be an identity element, (2) every element must have an inverse, and (3) the associative law must be obeyed.
		A subset of a group which is itself a group with respect to the same operation is called a subgroup.
		Basically, the center of a group is the collection of elements in the group that commute with all elements in the group and the centralizer of a given element in the group is the collection of all elements in the group that commute with that given element.
	libraryofmath.com /Group_Theory.html (1788 words)

	Group Theory and Physics
		Quantum mechanics showed that the elementary systems that matter is made of, such as electrons and protons, are truly identical, not just very similar, so that symmetry in their arrangement is exact, not approximate as in the macroscopic world.
		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.
	www.du.edu /~jcalvert/phys/groups.htm (5735 words)

	Springer Online Reference Works (Site not responding. Last check: 2007-11-05)
		The origin of proof theory can be traced to Antiquity (the deductive method of reasoning in elementary geometry, Aristotelian syllogistics, etc.), but the modern stage in its development begins at the turn of the 19th century with the studies of G.
		Proof theory disposes of powerful methods for the interpretation of theories in terms of other theories; such interpretations may also be used to establish the undecidability of several very simple calculi in which recursive calculi are not interpreted directly.
		Examples are elementary group theory, the theory of two equivalence relations, an elementary theory of fractional order, etc. On the other hand, examples are available of interesting decidable theories such as elementary geometry, the elementary theory of real numbers, and the theory of sets of natural numbers with a unique successor operation.
	eom.springer.de /p/p075430.htm (2500 words)

	Fundamental Theory Group: Elementary Particles & Fields (Site not responding. Last check: 2007-11-05)
		Field theory is central to understanding the extreme quantum and relativistic phenomena at this level of structure.
		In fact, the general approach seems to have become a paradigm for treating theories in which either the exact behavior is too difficult to obtain or the theory, away from a low energy region, is actually unknown.
		Their first study, dealing with abelian theories demonstrated that all spontaneously broken abelian supersymmetric theories admit cosmic string solutions which are superconducting due to fermion zero modes.
	www-hl.syr.edu /depts/Physics/FTGElementary.htm (1744 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 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.
		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.
	mathforum.org /library/topics/group_theory (2240 words)

	Using Maple V in Introductory Group Theory (Site not responding. Last check: 2007-11-05)
		Another type of assignments requires students to identify a group by means of a set of generators and a set of relations between those generators.
		Embedded Subgroups of a Symmetric Group and Cayley's Theorem
		By comparing an embedded subgroup of a symmetric group with a symmetric group of lower degree having exactly the same elements (in disjoint cycle notations), the students will see the distinction between subgroup of a symmetric group and a subset which is also a group but not a subgroup of that symmetric group.
	www.atcminc.com /mPublications/EP/EPATCM99/ATCMP002/PAPER/paper.html (1881 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)

	Elliptic Curves and Elliptic Functions
		When these varieties also have a group operation (that is regular as a mapping of varieties), it is called an algebraic group.
		The first is the group of all points on the curve E which have an order that divides m for some particular integer m.
		One of the principal facts of elementary group theory is that any finitely generated abelian group is the direct sum of a finite group and a finite number of infinite cyclic groups (isomorphic to the integers Z).
	cgd.best.vwh.net /home/flt/flt03.htm (3513 words)

	normal
		The group of all rotational symmetries of the cube such that the axis of rotation either passes through the center of 2 opposing faces or through 2 opposing vertices.
		For example, the group of 4 rotations of a cube along the x-axis basically "looks the same" as the group of 4 rotations of a cube along the y-axis, or along the z-axis for that matter.
		Groups: 1) group of rotational symmetries of a tetrahedron 2) group of rotational symmetries of a cube (which is effectively the same as that of an octahedron) 3) group of rotational symmetries of an icosahedron (which is effectively the same as that of a dodecahedron)
	math.ucr.edu /home/baez/normal.html (2662 words)

	BEACHY / BLAIR: ABSTRACT ALGEBRA
		For example, cyclic groups are introduced in Chapter 1 in the context of number theory, and permutations are studied in Chapter 2, before abstract groups are introduced in Chapter 3.
		One problem with most treatments of abstract algebra, whether they begin with group theory or ring theory, is that the students simultaneously encounter for the first time both abstract mathematics and the requirement that they produce proofs of their own devising.
		In Chapter 3 the abstract definition of a group is introduced, and the students encounter the notion of a group armed with a variety of concrete examples.
	www.math.niu.edu /~beachy/abstract_algebra_2ed/index.html (1861 words)

	Group Analyzer page
		Group Analyzer is a program for students and instructors of elementary group theory.
		However, the groups are of "intermediate" size, of order up to 400, and are big enough to illustrate various ideas of finite groups in non-trivial ways.
		I have tried to design Group Analyzer so that it will be easy for students to use (and learn to use) and not become a distraction from the main task of learning group theory.
	faculty.cua.edu /glenn/gapageOLD.html (782 words)

	HBA_Linear Algebra and Group Theory for Physicists_ KNS RAO
		Professor Srinivasa Rao's text on Linear Algebra and Group Theory is directed to undergraduate and graduate students who wish to acquire a solid theoretical foundation in these mathematical topics which find extensive use in physics.
		An authority on diverse aspects of mathematical physics, Professor K N Srinivasa Rao taught at the University of Mysore until 1982 and was subsequently at the Indian Institute of Science, Bangalore.
		The first edition of Linear Algebra and Group Theory for Physicists was co-published in 1996 by New Age International, and Wiley, New York.
	www.hindbook.com /Home.asp?P=131 (186 words)

	Advanced Algebra
		Text: Concerning group theory we shall use D. Robinson, A Course in the Theory of Groups and depending on the pace of the course we might cover also parts of M. Isaacs, Character Theory of Finite Groups.
		In particular, you should be familiar with the following concepts and theorems in group theory: group, subgroup, order of an element, cyclic group, Lagrange's theorem, homomorphism, normal subgroup, factor group, homomorphism and isomorphism theorems, symmetric and alternating groups, direct product.
		If all proper subgroups of a finite group G are Abelian, then G is solvable.
	www.stolaf.edu /depts/math/budapest/WebPages/course_AAL.html (279 words)

	Conley: 3520 Spring 2004 (Site not responding. Last check: 2007-11-05)
		You will need to know elementary group theory and elementary ring theory: familiarity with groups of small order and the ring Z_n of integers modulo n will be sufficient.
		The unit Hamiltonian sphere is a multiplicative group isomorphic to a group you may have heard of: the matrix group SU_2.
		Galois theory, coupled with the structure of the symmetric groups S_3 and S_4, easily yield the formulae for the solution of the general cubic and quartic equations.
	www.math.unt.edu /~conley/3520s06.htm (463 words)

	Amazon.com: Groups and Symmetry (Undergraduate Texts in Mathematics): Books: M. A. Armstrong (Site not responding. Last check: 2007-11-05)
		The theory is amplified, exemplified and properly related to what this part of algebra is really for by discussion of a wide variety of geometrical phenomena in which groups measure symmetry.
		Throughout the book, emphasis is placed on concrete examples, often geometrical in nature, so that finite rotation groups and the 17 wallpaper groups are treated in detail alongside theoretical results such as Lagrange's theorem, the Sylow theorems, and the classification theorem for finitely generated abelian groups.
		The discussion of the symmetry groups of Platonic solids is both enjoyable in itself and useful for visualizing groups.
	www.amazon.com /Groups-Symmetry-Undergraduate-Texts-Mathematics/dp/0387966757 (1461 words)

	Albright College - Academics: Mathematics
		The mathematics courses are designed to provide a thorough undergraduate training in mathematics for those students who wish to pursue graduate study in the subject, teach mathematics in the secondary or elementary schools, or to work in various fields in business and industry.
		Students concentrating in mathematics are required to complete MAT 107, 108, 207, 307, 308 or 312, 311, 491, and 492; three mathematics courses at the 300 level to be chosen with departmental approval; and PHY 201 and 202.
		Elementary analytic geometry will be discussed, along with the algebra and composition of functions, inverse functions, trigonometry, and logarithmic and exponential functions.
	www.albright.edu /academics/depts/math.html (1163 words)

	Elementary group theory - ExampleProblems.com
		Theorem 1.6: For all a belonging to a group (G,*), (a
		One handy theorem that covers the case for both finite and infinite groups is:
		Theorem 2.5: Let a be an element of a group (G,*).
	www.exampleproblems.com /wiki/index.php/Elementary_group_theory (1692 words)

	HR - Automatic Theory Formation In Pure Mathematics
		Please note that the results here are out of date, and we are planning another evaluation of HR in group theory.
		We wanted HR to invent a calculation which is an invariant (so it gives the same results for any pair of isomorphic groups), and which classifies the groups up to six (so it gives different results for any pair of non-isomorphic groups).
		As a tricky example, we wanted HR to find a property of the groups D(2) and D(3) which is not shared by any other group up to order six.
	www.dai.ed.ac.uk /homes/simonco/research/hr/group_theory.html (559 words)

	Peg Solitaire and Group Theory
		Not long ago, with the help of very elementary group theory, Arie Bialostocki from University of Idaho proved that there are only five locations (b.
		The irony is in that from the same position the player can leave the sole remaining peg in the central hole, thus gaining the status of genius, instead of an outstanding player.
		Bialostocki, An Application of Elementary Group Theory to Central Solitaire, The College Mathematics Journal, v 29, n 3, May 1998, 208-212.
	www.cut-the-knot.org /proofs/PegsAndGroups.shtml (793 words)

	MUG: group theory (4.11.98) (Site not responding. Last check: 2007-11-05)
		I would like to use Maple in teaching elementary group theory and have a number of questions.
		They are very elementary, but illustrate some of the things you are interested in for undergraduate modern algebra students.
		If memory serves me right Maple's groups package is pretty limited and not very impressive for doing even elementary group theory.
	www.math.rwth-aachen.de /mapleAnswers/html/648.html (162 words)

	Discrete Mathematics Graduate Courses (Site not responding. Last check: 2007-11-05)
		Elementary number theory, polynomial and abstract rings, ideals and quotient rings, PID's and Eclidean rings, groups, cyclic and Abelian groups, direct products, algebraic field extensions, splitting fields, field automorphisms, finite fields.
		MATH-689 Combinatorics and Graph Theory II Methods of linear algebra in combinatorics and graph theory, basics of coding theory and associated designs, number theory and cryptography, communication complexity.
		An introduction to large deviation theory and martingale theory.
	www.math.udel.edu /research/DiscreteMath/courses.html (241 words)

	Elementary Number Theory, Group Theory and Ramanujan Graphs - Cambridge University Press
		Expander graphs are both highly connected but sparse, and besides their interest within combinatorics and graph theory, they also find various applications in computer science and engineering.
		The reader needs only a background in elementary algebra, analysis and combinatorics; the authors supply the necessary background material from graph theory, number theory, group theory and representation theory.
		The text can therefore be used as a brief introduction to these subjects as well as an illustration of how such topics are synthesised in modern mathematics.
	www.cambridge.org /catalogue/catalogue.asp?ISBN=0521824265 (284 words)

	Evariste Galois . . . group theory in elementary terms (Site not responding. Last check: 2007-11-05)
		I believe it was in the field of group theory.
		Response #: 1 of 1 Author: asmith Actually, Galois sort of invented group theory while trying to solve this problem - the problem was, can you find a formula like the famous quadratic formula that finds the roots of a fifth degree polynomial?
		Formulas were known at this time for all polynomials of degree 3 or four, but there was no general method for finding roots of higher order polynomials.
	www.newton.dep.anl.gov /newton/askasci/1995/math/MATH119.HTM (279 words)

	Group theory with MAPLE (Site not responding. Last check: 2007-11-05)
		This page describes a small file with some elementary group theory commands in MAPLE which may be useful to help in teaching a course in group theory.
		find the group table of a permgroup (both as a table of permutations and as a more compact looking table of indexed letters),
		Other MAPLE packages which use group theory are ACE, the MAPLE packages group and combinat, and the MAPLEV5 share packages perm and coxeter.
	cadigweb.ew.usna.edu /~wdj/maplestuff/group.html (316 words)

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