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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.
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.
The same Cayley table of S 3 ... The symmetric group S 3 as a subgroup of S 4. See: Subgroups of S 4. There is also: left action last ⋅ first: This is: right action
Description: Cayley graph of S 4. Red arrows stand for permutation number 9, moving all elements one place to the right, and the rightmost element on the leftmost place.Blue arrows stand for permutation number 4, which is a similar move to le left, but leaving the rightmost element unchanged.
This is the corresponding graph, called the Nauru graph: The red, green and blue squares form the permutation matrices, indexed 1, 5 and 21 in the following file. (Compare: v:Symmetric_group_S4#A_closer_look_at_the_Cayley_table) 1, 5 and 21 are the generators of the Nauru graph. Source: Own work: Author
Cayley graph of S 4 generated by the transpositions that swap neighbouring elements (a truncated octahedron like the permutohedron) This image was created with POV-Ray.
Description: Cayley graph of S 4, . generated by blue: (12) green: (13) red: (14) This is a Nauru graph, compare The many faces of the Nauru graph by 0xDE. The colors of the vertices are like these , and have nothing to do with the colors of the edges.
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