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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.
One sheet of the Cayley graph of the Baumslag–Solitar group BS(1, 2). Red edges correspond to a and blue edges correspond to b. The sheets of the Cayley graph of the Baumslag-Solitar group BS(1, 2) fit together into an infinite binary tree. Visualization comparing the sheet and the binary tree Cayley graph of (,).
This is usually done by studying the Cayley graphs of groups, which, in addition to the graph structure, are endowed with the structure of a metric space, given by the so-called word metric. Geometric group theory, as a distinct area, is relatively new, and became a clearly identifiable branch of mathematics in the late 1980s and early 1990s.
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The same file with right action (which is more usual for Cayley graphs). One of the Cayley graphs of the dihedral group Dih 4. This version of File:Dih 4 Cayley Graph; generators a, b.svg uses prefix notation, which is unusual for Cayley graphs. In this file an arrow for s goes from g to sg, while in the other file it goes from g to gs.
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(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)