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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.
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.
Hence, the fundamental group of the Cayley graph Γ(G) is isomorphic to the kernel of φ, the normal subgroup of relations among the generators of G. The extreme case is when G = {e}, the trivial group, considered with as many generators as F, all of them trivial; the Cayley graph Γ(G) is a bouquet of circles, and its fundamental group is F ...
Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations.
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 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.
In mathematics, Cayley's formula is a result in graph theory named after Arthur Cayley. It states that for every positive integer n {\displaystyle n} , the number of trees on n {\displaystyle n} labeled vertices is n n − 2 {\displaystyle n^{n-2}} .
Cayley table as general (and special) linear group GL(2, 2) In mathematics, D 3 (sometimes alternatively denoted by D 6) is the dihedral group of degree 3 and order 6. It equals the symmetric group S 3. It is also the smallest non-abelian group. [1] This page illustrates many group concepts using this group as example.