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Cayley tables were first presented in Cayley's 1854 paper, "On The Theory of Groups, as depending on the symbolic equation θ n = 1". In that paper they were referred to simply as tables, and were merely illustrative – they came to be known as Cayley tables later on, in honour of their creator.
Thus, normalizing a Cayley table (putting the border headings in some fixed predetermined order by permuting rows and columns including the headings) preserves the isotopy class of the associated Latin square. Furthermore, if two normalized Cayley tables represent isomorphic quasigroups then their associated Latin squares are also isomorphic.
Cayley table of the symmetric group S 4 The odd permutations are colored: Transpositions in green and 4-cycles in orange Cayley table of the alternating group A 4 Elements: The even permutations (the identity, eight 3-cycles and three double-transpositions (double transpositions in boldface)) Subgroups:
The Cayley table of the group can be derived from the group presentation , = =, = . A different Cayley graph of D 4 {\displaystyle D_{4}} is shown on the right. b {\displaystyle b} is still the horizontal reflection and is represented by blue lines, and c {\displaystyle c} is a diagonal reflection and is represented by pink lines.
Arthur Cayley FRS (/ ˈ k eɪ l i /; 16 August 1821 – 26 January 1895) was a British mathematician who worked mostly on algebra. He helped found the modern British school of pure mathematics , and was a professor at Trinity College, Cambridge for 35 years.
A Cayley graph of the symmetric group S 4 using the generators (red) a right circular shift of all four set elements, and (blue) a left circular shift of the first three set elements. Cayley table, with header omitted, of the symmetric group S 3. The elements are represented as matrices. To the left of the matrices, are their two-line form.
Cayley table as general (and special) linear group GL(2, 2) In mathematics , D 3 (sometimes alternatively denoted by D 6 ) is the dihedral group of degree 3 and order 6. It equals the symmetric group S 3 .
A Cayley table lists the results of all such compositions possible. For example, rotating by 270° clockwise ( r 3 {\displaystyle r_{3}} ) and then reflecting horizontally ( f h {\displaystyle f_{\mathrm {h} }} ) is the same as performing a reflection along the diagonal ( f d {\displaystyle f_{\mathrm {d} }} ).