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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.
Because the cancellation property holds for groups (and indeed even quasigroups), no row or column of a Cayley table may contain the same element twice. Thus each row and column of the table is a permutation of all the elements in the group. This greatly restricts which Cayley tables could conceivably define a valid group operation.
(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)
In mathematics, a Cayley graph, also known as a Cayley color graph, Cayley diagram, group diagram, or color group, [1] is a graph that encodes the abstract structure of a group. Its definition is suggested by Cayley's theorem (named after Arthur Cayley ), and uses a specified set of generators for the group.
The same Cayley table of S 3 with the permutations represented by 3x3 matrices Positions in the Cayley table. The symmetric group S 3 as a subgroup of S 4. See ...
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Cayley table of the general linear group GL(2,2) which is isomorphic to S 3. The determinant of these six matrices is 1 (as there is no other nonzero element in F 2), so this is also the special linear group SL(2,2).
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.