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Topic: Cayley table

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	PlanetMath: Cayley table
		A Cayley table for a group is essentially the “multiplication table” of the group.
		The columns and rows of the table (or matrix) are labeled with the elements of the group, and the cells represent the result of applying the group operation to the row-th and column-th elements.
		All Cayley tables for isomorphic groups are isomorphic (that is, the same, invariant of the labeling and ordering of group elements).
	planetmath.org /encyclopedia/CayleyTable.html (203 words)

	Arthur Cayley Summary
		Cayley was born in England during a short visit of his parents to Richmond, Surrey, but spent his first eight years in Russia where his father plied a living as a merchant.
		Cayley was reluctant to take holy orders, however, and in light of the regulations of the time he was required to give up his fellowship after a certain period.
		Cayley entered Lincoln's Inn, one of the Inns of Court for the training of prospective lawyers in 1846 and was admitted to the bar in 1849.
	www.bookrags.com /Arthur_Cayley (4857 words)

	Cayley Graphs (Site not responding. Last check: 2007-10-04)
		Most Cayley graphs however cannot be realised in two or three dimensions with all its edges the same length.
		Cayley graphs are uniform, so this dangling string net will look exactly the same regardless of which node you picked it up at.
		In the general Cayley graph we might have two faces that share more than one edge, or an edge that is shared by more than two faces.
	www.geocities.com /jaapsch/puzzles/cayley.htm (4449 words)

	Cayley table - Wikipedia, the free encyclopedia
		In essence, the Cayley table is the group-theoretic analogue of an addition or a multiplication table.
		Cayley tables were presented in an 1889 paper of Cayley's, On The Theory of Groups, and focused on the construction of colourgroups, graphical depictions of groups by coloured lines and routes.
		Cayley originally set up his "squares" so that the identity element was first, giving rise to the first row and column acting as column headers.
	en.wikipedia.org /wiki/Cayley_table (1629 words)

	About Cayley Quilt Maker (Site not responding. Last check: 2007-10-04)
		The quilts are made using Cayley tables as a basic pattern which is repeatedly tiled across the quilt surface.
		The D4 Table quilt is a Cayley table for the dihedral group D4, whose motions are applied to the image of the original table.
		The color of each pixel in the image of the table is increased by the sum of its x and y coordinates modulo n, where n is the size of the table's image in pixels.
	www.wou.edu /~burtonl/about.html (334 words)

	Cayley biography
		In 1863 Cayley was appointed Sadleirian professor of Pure Mathematics at Cambridge.
		He gives the 'Cayley tables' of some special permutation groups but, much more significantly for the introduction of the abstract group concept, he realised that matrices and quaternions were groups.
		Cayley developed the theory of algebraic invariance, and his development of n-dimensional geometry has been applied in physics to the study of the space-time continuum.
	www-history.mcs.st-andrews.ac.uk /Biographies/Cayley.html (1058 words)

	Finite Groups and Cayley Tables
		In any Cayley table of a finite group the identity elements must be distributed symmetrically about the main diagonal.
		Thus this cannot be the Cayley table for a group as in a group every right inverse is also a left inverse.
		It is not simply a matter of glancing at the Cayley table of a group to notice whether it is the same group as is represented in a different Cayley table.
	members.tripod.com /~dogschool/cayley.html (1289 words)

	Arthur Cayley (Site not responding. Last check: 2007-10-04)
		Arthur Cayley showed great skill in numerical calculations at school and his aptitude for advanced mathematics was apparent.
		He spent 14 years as a lawyer but Cayley, although very skilled in the legal specialty of conveyancing, always considered it as a means to make money so that he could pursue mathematics.
		In 1854, Cayley wrote two papers which are remarkable for the insight they have of abstract groups.
	www.stetson.edu /~efriedma/periodictable/html/C.html (553 words)

	Maths -Groups Cayley Table - Martin Baker
		We can use a table to represent a finite group, but we can also use a Cayley Table to represent the multiplication rules of a group whose elements consist of a number of scalar dimentions, the table tells us how these dimentions combine.
		The table only represents the type of each entry and whether there is any sign reversal, to actually multiply two whe have to multiply the scalar values to determine there contribution to the given dimention.
		For instance, imagine we have an algebra defined by table 1 and the elements of this algebra are another algebra defined by table 2.
	www.euclideanspace.com /maths/algebra/groups/cayley/index.htm (1154 words)

	Group theory terms
		A multiplication table for a group is named thus because it is much like elementary school multiplication tables, except that it uses the group elements and operation rather than integers under ordinary, everyday multiplication.
		Thus the table is a grid, and across the top row and down the left column every element of the group is listed, and filling the rest of the table is the results of applying the group operation to the elements in the header row and column.
		Multiplication tables and Cayley diagrams can both organize themselves by the cosets of a subgroup and then separate those cosets (or "chunk" them) to help you visualize the quotient operation.
	groupexplorer.sourceforge.net /help/rf-groupterms.html (3947 words)

	Cayley Tables and Quilts - The National Curve Bank: A Math Archive (Site not responding. Last check: 2007-10-04)
		If group G has 8 elements, then you will have 8 different pieces quilted together to form a 8 x 8 "square." Similarly, the Cayley table for this group will have 64 entries.
		Cayley tables are studied in an advanced topic of mathematics named Group Theory.
		The novice may also want to explore the connection between Cayley tables, quilt patterns and modular, or "clock" arithmetic.
	curvebank.calstatela.edu /quilt/quilt.htm (743 words)

	Permutation Groups
		With this convention decided upon together with the convention for Cayley tables that left element in the operation is in the column to the left of the table and the right element is in the row along the top of the table we get the following table for S
		It's exactly the same table as we gave for the Symmetry Group of the Equilateral Triangle.
		Cayley's Theorem: Every Group is isomorphic to a group of permutations.
	members.tripod.com /~dogschool/permgroups.html (2347 words)

	Bol Loops of Order 16
		In some cases these invariants (and those displayed in the linked pages available) are insufficient to distinguish isomorphism classes; see comments accompanying one of our tables.
		Cayley tables for all 2052 Bol loops of order 16: zipped text file (230 KB) or gzipped text file (178 KB)
		Entries in the tables display only the value of k when the other parameters i,c are specified by column headings; and links are provided to pages describing the corresponding loops.
	www.uwyo.edu /moorhouse/pub/bol/bol16.html (923 words)

	Cayley Tables (Site not responding. Last check: 2007-10-04)
		A Cayley Table is really just like the square tables for multiplication or addition that one sees in primary school, but generalised to abstract algebraic structures.
		At this stage, this is nothing more than just a set of symbols, two of which happen to be symbols we frequently use to represent the number one and the number zero (or whatever zero is if it isn't a number).
		Aside from the historical significance, this is a bloody fantastic table, with plenty of zero's in it.
	www.maths.utas.edu.au /People/Jackson/cayley.html (1134 words)

	\color{red}\Large{\textbf{2GA2\quad 2000\quad 539.231}} (Site not responding. Last check: 2007-10-04)
		Prove that, in each column of a Cayley table for a finite group (G, *), each element of G occurs exactly once.
		To prove that this group is nonabelian, we choose from the Cayley table any pair of elements which do not commute, for example, a
		= 25 = 1, it follows that the second Cayley* table is the Cayley table for a cyclic group, and that 2 is a generator.
	www.maths.uwa.edu.au /~csaba/2GA2/exsolns6.html (317 words)

	Moduloid - Abelian Unital Magma
		From the above expression, the Cayley table is composed as,
		In this case the Cayley table for such group is expressed as,
		On the table the unit elements (0, 0) are specially indicated with red.
	geocities.com /tontokohirorin/mathematics/moduloid/moduloid2.htm (1336 words)

	Directions for Use -- The Abstract Algebra Helper
		Cayley button requires the group radio button to be selected before the cayley table button is pressed.
		Cayley button requres the "n" value for the group to be entered before the cayley table button is pressed.
		Generator button requires the group to be selected (the radio button is filled), the "n" value for the group to be entered, and an element of the group to be entered.
	www.stolaf.edu /people/dietz/AlgHelper/directions.html (598 words)

	Sudoku Text - Physics Forums Library
		10-25-2005, 05:14 AM A Latin square or a Cayley table is a table which defines what a combination of 2 group elements would give.
		10-25-2005, 08:54 AM A Latin square or a Cayley table is a table which defines what a combination of 2 group elements would give.
		You can use an n by n Latin square to define a binary operation on a set of n elements, but you won't expect the group properties to hold except of course closure (it's a quasigroup).
	www.physicsforums.com /archive/index.php/t-96311.html (697 words)

	Real Cayley, quaternion, and octonian algebras (Site not responding. Last check: 2007-10-04)
		(elements of the Cayley algebra associated to R^3).
		By the well-known polarization identity, one may create out of this norm a bilinear form on the Cayley numbers, given below.
		The octonian algebra is isomorphic to the Cayley algebra.
	web.usna.navy.mil /~wdj/cayley0.htm (313 words)

	Oakland Table: Speakers: David Cayley (Site not responding. Last check: 2007-10-04)
		David Cayley is a writer and broadcaster who has been working for the national radio network of the Canadian Broadcasting Corporation for nearly thirty years.
		He is the author of five books, including Ivan Illich in Conversation, and most recently, The Expanding Prison, an essay on why there are so many people in prison and on alternatives ways of addressing crime and punishment.
		In 1989 Cayley presented a five-hour radio series called "Part Moon, Part Travelling Salesman: Conversations with Ivan Illich." In January of this year he produced a new series called "The Corruption of Christianity: Ivan Illich on Gospel, Church and Society."
	www.wtp.org /bios/cayleybio.html (143 words)

	Card Colm - Martin Gardner's Magic Spells (Site not responding. Last check: 2007-10-04)
		That was a relevant anagram for a recasting of the replacement rules--it's the (Arthur) Cayley table for any two-element group (with D as the neutral element).
		These are two manifestations of the Cayley table for a three-element group (with D as the neutral element).
		Since 1988, Colm Mulcahy (colm@spelman.edu) has been in the department of mathematics at Spelman College, where a recent project of his was the creation and launching of the new BIO SIGMAA webpage.
	enterprise.maa.org /columns/colm/cardcolm200612.html (1941 words)

	groupMatrix
		Stores the Cayley table as an ArrayList of ArrayList of Strings where each row in the table is one of the ArrayList of Strings and each cell is represented by one String.
		The class also has analysis functions to determine if the Cayley Table has group characteristics such as an Identity element, inverse for each element, and associativity.
		ArrayList of ArrayList of Strings used for storing the Cayley Table where each row in the table is one of the ArrayList of Strings and each cell is represented by one String.
	www.groovypower.com /thesis/SampleCode/javadoc/cayleytable/groupMatrix.html (762 words)

	Fictionally Speaking (Site not responding. Last check: 2007-10-04)
		However, not all Cayley tables will have each element in both diagonals.
		For this problem I started with the first row of the Cayley table as (from left to right) 0,1,2,3,4.
		Using these rules, there are two Cayley tables which will yield appropriate solutions.
	www.boomspeed.com /lisaqt3/Trees.html (245 words)

	[No title]
		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.
		So one of our projects is to have the students write a procedure to construct a table for a permutation group G. The procedure calls the already defined function gpElements to create a list of all elements of G then it constructs a two dimensional array of size o(G) x o(G).
		Since the abstract group Table for G, will be the same as the abstract group table for PG we can use our procedure abs_gp_table(n,PG) to construct an abstract group table for G. Cayley's Table: an application of the procedure abs_gp_table
	archives.math.utk.edu /ICTCM/EP-11/C23/paper.html (1694 words)

	Cayley, Arthur - HighBeam Encyclopedia (Site not responding. Last check: 2007-10-04)
		Cayley's collected papers were published in 13 volumes (1889-98).
		Find newspaper and magazine articles plus images and maps related to "Cayley, Arthur" at HighBeam.
		More information is at your fingertips at HighBeam Research:
	www.encyclopedia.com /doc/1E1-cayley-a.html (229 words)

	Abstract_Algebra.nb
		You will see that the two Cayley Tables do not appear to be Isomorphic, but if the order of terms is changed in Z[4], you will quickly see the isomorphisms.
		When I want to make a Cayley Table of my new group, I can call it G3.
		Now, if you compare the Cayley table of U[10] with that of G3, (which is actually Z[4]), you will see that their color schemes match, and that they are isomorphic.
	www.unl.edu /tcweb/fowler/890Stuff/nikiGroup (238 words)

	MAT 313 Homework
		Cayley Table for Z[4] Cayley Table for Z[2](+)Z[2]
		a) The Cayley table for the quaternion group G is printed below.
		c) The Cayley table for G/H, where H={1,-1}, is printed below.
	www.math.sunysb.edu /~joa/MAT313/hw-VIII---313.html (80 words)

	Peg Solitaire and Group Theory (Site not responding. Last check: 2007-10-04)
		to indicate the fact expressed in the first row of the table, namely, that whenever peg x jumps over peg y it always lands in hole z.
		Similar notion are used for the remaining rows of the table, so that, for example,
		The Cayley table of a group collects all the information about the group operation ("+" in our case) in compact form.
	www.cut-the-knot.org /proofs/PegsAndGroups.shtml (793 words)

	Cayley graph - Wikipedia, the free encyclopedia
		If the set S doesn't generate the whole group, the Cayley graph isn't connected.
		Since the free group has no relations, the graph has no cycles.
		The Sabidussi theorem gives a characterization of Cayley graphs: Graph X is a Cayley graph if and only if the automorphism group of X contains a subgroup G acting regularly on the vertex set of X.
	en.wikipedia.org /wiki/Cayley_graph (620 words)

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