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Topic: Stabilizer (group theory)

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	Group Theory for puzzles
		The tetrahedral group, the symmetry of a tetrahedron.
		The octahedral group, the symmetry of a cube or octahedron.
		The icosahedral group, the symmetry of a dodecahedron or icosahedron.
	www.geocities.com /jaapsch/puzzles/groups.htm (9058 words)

	GAP Manual: 21.6. Stabilizer Chains
		The chain of subgroups of G itself is called the stabilizer chain of G relative to B.
		For permutation groups of small degree this might be possible, but for permutation groups of large degree it is still not good enough.
		Note that the records that represent the stabilizers are not group records (see Group Records).
	www.math.uiuc.edu /Software/GAP-Manual/Stabilizer_Chains.html (658 words)

	MTH-3D15 : Theory of Finite Groups (Site not responding. Last check: 2007-10-08)
		Group theory is a large topic which interconnects with many branches of pure and applied mathematics.
		Overview: Group Theory has two main roots, one in geometry where groups of geometrical transformations were studied, the other in algebra and the theory of equations where groups of substitutions of variables (i.e.
		Abstract groups began to emerge with Jordan's seminal Traité des substitutions et des equations algébriques (1870) while the definition of abstract groups in general appears to be due to Weber (1882).
	www.mth.uea.ac.uk /maths/syllabuses/0506/3D1505.html (693 words)

	MC341 Group Theory (Site not responding. Last check: 2007-10-08)
		One of the module's major goals is to develop enough theory to be able to discuss in a worthwhile fashion the classification of the finite simple groups.
		Soluble groups, commutator subgroups, examples, central series and nilpotent groups, p-groups are nilpotent, maximal subgroups of nilpotent groups are normal, solubility of groups of order less than 60.
		J.J. Rotman, An Introduction to the Theory of Groups, Springer-Verlag.
	www.mcs.le.ac.uk /Modules/Year3_98-99/MC341.html (709 words)

	William M. Kantor: Research
		The 2-transitive permutation representations of the finite Chevalley groups.
		Groups and Geometry in Honor of R. Bruck, Algebras, Groups and Geometries 2 (1985) 313-322.
		Theory (A) 113 (2006) 1594--1613 (with B. van Asch, A. Blokhuis, H. Hollmann, and H. van Tilborg).
	darkwing.uoregon.edu /~kantor/research.html (1254 words)

	Group action - Wikipedia, the free encyclopedia (via CobWeb/3.1 planet03.csc.ncsu.edu) (Site not responding. Last check: 2007-10-08)
		In this case, the group is also called a permutation group (especially if the set is finite or not a vector space) or transformation group (especially if the set is a vector space and the group acts like linear transformations of the set).
		The automorphism group of a vector space (or graph, or group, or ring...) acts on the vector space (or set of vertices of the graph, or group, or ring...).
		A group action is then nothing but a functor from G to the category of sets, and a group representation is a functor from G to the category of vector spaces.
	en.wikipedia.org.cob-web.org:8888 /wiki/Group_action (2407 words)

	Group Orbit -- from Wolfram MathWorld
		In celestial mechanics, the fixed path a planet traces as it moves around the sun is called an orbit.
		A group fixed point is an orbit consisting of a single element, i.e., an element that is sent to itself under all elements of the group.
		The equator is a one-dimensional orbit, as is a general orbit, corresponding to a line of latitude.
	mathworld.wolfram.com /GroupOrbit.html (274 words)

	Group Theory book
		It is kind of fun to decompose, for each n, the group of units of Z(n), into a direct sum of cyclic groups.
		the first interesting ones are "dihedral" groups, (symmetries of a polygon), then the platonic solid groups, symmetries of the cube, tetrahedron, and (the first really interesting one) the "icosahedral group", symmetries of the icosahedron, of order 60, and isomorphic to the alternating group A(5).
		Of course one should be aware of the symmetric groups S(n), of all permutations of a set of n elements, in which A(n) is the unique normal subgroup of index 2.
	www.physicsforums.com /showthread.php?p=920330 (1133 words)

	Music and Mathematics
		The theories of semigroups and monoids are not simpler than the theory of groups.
		A group action of a group G (with binary operation ) on set A is a map from G X A to A, written as g(a) for all g in G and a in A, such that universally in G and A, g1(g2(a))=(g1g2)(a) and e(a)=a (e the identity in G).
		The quotient group G/H: the set of fibers over elements of H a subgroup of G can be a group with the binary operation defined by XaXb=Xab (the combination of the fibers over a and b is the fiber over the combination (in H) of a and b).
	faculty.washington.edu /jrahn/5752004.htm (5226 words)

	Dodecahedral Faces of the Mathieu group of degree 12
		Groups are objects in mathematics that measure symmetry in nature.
		A group is a set with a binary operation that has an inverse, an identity and is associative.
		Robinson, A Course in the Theory of Groups, Springer, 1996.
	web.usna.navy.mil /~wdj/m_12.htm (3203 words)

	Films Media Group - Inversive Geometry
		Examing the idea of isomorphism in group theory, this program shows how two groups may really be the same, even though they look totally different at first sight.
		This program explores the orbit-stabilizer theorem in group theory, showing how this leads to the counting theorem, which is used to solve counting problems.
		Films Media Group, Films for the Humanities and Sciences, Cambridge Educational, Meridian Education, Shopware and their respective logos are trademarks of Films Media Group, a PRIMEDIA company.
	www.films.com /id/8605/Inversive_Geometry.htm (220 words)

	MA30110 - GROUP THEORY
		The concept of a group occurs naturally in situations involving symmetry or in which some quantity is being preserved; for example, various letters such as A, S and I possess different numbers of symmetries and rigid motions preserve distance.
		The principal structure theorems for finite groups will be described and applied in a variety of group theoretic contexts.
		To provide a deeper understanding of the concepts and techniques of abstract algebra, introduced in module MA20310, by focusing on the group concept, starting with an axiomatic development of group theory, establishing a structure theory, mainly in the context of finite groups, and giving brief illustrations of a selection of applications of group theory.
	www.aber.ac.uk /modules/current/MA30110.html (343 words)

	Fall 2006 New York Group Theory Seminar (via CobWeb/3.1 planet03.csc.ncsu.edu) (Site not responding. Last check: 2007-10-08)
		If the condition on the action is weakened to allow infinite stabilizers, we instead obtain that the group is weakly relatively hyperbolic with respect to a finite set of stabilizer subgroups.
		We apply this to prove that many Artin groups are weakly hyperbolic relative to their spherical parabolics by showing that the Deligne complex for these Artin groups supports a CAT(-1) metric.
		The New York Group Theory Seminar and some of the associated conferences are supported by funds from the National Science Foundation, Dean of Science, Maria Tamargo and Dean of Engineering, Joe Barba.
	zebra.sci.ccny.cuny.edu.cob-web.org:8888 /web/grouptheory.org/CharneyAbstract.html (170 words)

	Teaching page of Alexander Yong (Site not responding. Last check: 2007-10-08)
		Cyclic groups, any subgroup of a cyclic group is cyclic (idea of proof: in the end, consider the division algorithm).
		The kernel of Phi is Stab_{G}(X), the stabilizer subgroup of X. This kernel is {e} iff Phi is 1-1 iff Stab_{G}(X)={e} iff G's action on X is "faithful".
		Then explained the ring theory analogues of homomorphism theorems: an subset S of a ring R is an ideal if and only if it is the kernel of a ring homomorphism and if and only if R/S is a ring.
	math.berkeley.edu /~ayong/teaching_Math113_Fall2003.html (2733 words)

	Stabilizer -- from Wolfram MathWorld
		For example, the stabilizers of 1 and 2 under the permutation group
		and the stabilizers of 3 and 4 are
		is transitive, and so is related to its isotropy group.
	mathworld.wolfram.com /Stabilizer.html (96 words)

	algsyl.html
		Examples include the cyclic, symmetric, alternating, dihedral groups, and groups of symmetries of the Platonic solids.
		The length of an orbit equals the index of the stabilizer.
		If n divides the order of the group, then the number of elements whose order divides n is a multiple on n.
	www.pitt.edu /~gmc/algsyl.html (308 words)

	Logic Colloquium 2006 (via CobWeb/3.1 planet03.csc.ncsu.edu) (Site not responding. Last check: 2007-10-08)
		This theory is founded on the construction of a well-ordering on the real numbers set with a partition in an infinity of subsets.
		Thinking of topoi as higher-order theories, this amounts to constructing, for any higher-order theory, a set theoretic universe incorporating the types of that theory as sets, in such a way that the set theory of the universe is a conservative extension of the higher-order theory we started out with.
		The study of multi-dimensional amalgamation properties in simple theories is motivated by the question of whether or not a model of a theory fails to interpret a cycle-free hypergraph.
	www.cs.ru.nl.cob-web.org:8888 /lc2006/contributed.html (10915 words)

	AMERICAN MATHEMATICAL MONTHLY - December 2001
		We describe how to use stabilizer codes to correct errors in a exposition assumes only a knowledge of linear algebra and a little bit of group theory.
		We begin with a brief primer of quantum theory and then describe qubit errors and the error group they generate.
		The error group has an associated geometry via the theory of extra special p-groups, developed in the 1950s by Philip Hall and Graham Higman.
	www.maa.org /pubs/monthly_dec01_toc.html (610 words)

	List of group theory topics - Wikipedia, the free encyclopedia
		See also: List of abstract algebra topics, List of category theory topics, list of Lie group topics.
		[edit] Mathematical objects which have (or make use of) a group operation
		See also list of harmonic analysis and representation theory topics
	en.wikipedia.org /wiki/List_of_group_theory_topics (198 words)

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