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The Cayley table tells us whether a group is abelian. Because the group operation of an abelian group is commutative, a group is abelian if and only if its Cayley table's values are symmetric along its diagonal axis. The group {1, −1} above and the cyclic group of order 3 under ordinary multiplication are both examples of abelian groups, and ...
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.
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.
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} }} ).
Cayley table of GL(2, 2), which is isomorphic to S 3. If F is a finite field with q elements, then we sometimes write GL(n, q) instead of GL(n, F). When p is prime, GL(n, p) is the outer automorphism group of the group Z p n, and also the automorphism group, because Z p n is abelian, so the inner automorphism group is trivial. The order of GL(n ...
(The generators a and b are the same as in the Cayley graph shown above.) Cayley table as multiplication table of the permutation matrices Positions of the six elements in the Cayley table Only the neutral elements are symmetric to the main diagonal, so this group is not abelian. Cayley table as general (and special) linear group GL(2, 2)
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: