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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.
The free group F S with free generating set S can be constructed as follows. S is a set of symbols, and we suppose for every s in S there is a corresponding "inverse" symbol, s −1, in a set S −1. Let T = S ∪ S −1, and define a word in S to be any written product of elements of T. That is, a word in S is an element of the monoid ...
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 (,).
Download QR code; Print/export Download as PDF; Printable version; In other projects ... Pages in category "Cayley graphs" The following 3 pages are in this category ...
Geometric group theory grew out of combinatorial group theory that largely studied properties of discrete groups via analyzing group presentations, which describe groups as quotients of free groups; this field was first systematically studied by Walther von Dyck, student of Felix Klein, in the early 1880s, [2] while an early form is found in the 1856 icosian calculus of William Rowan Hamilton ...
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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.
(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)