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Visualization comparing the sheet and the binary tree Cayley graph of (,). Red and blue edges correspond to a {\displaystyle a} and b {\displaystyle b} , respectively. In the mathematical field of group theory , the Baumslag–Solitar groups are examples of two-generator one-relator groups that play an important role in combinatorial group ...
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 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.
Description: Cayley table of Dih 4 (right action). One of the Cayley graphs of the dihedral group Dih 4. The red arrow represents permutation =, and the blue edge represents permutation =.
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A presentation of a group determines a geometry, in the sense of geometric group theory: one has the Cayley graph, which has a metric, called the word metric. These are also two resulting orders, the weak order and the Bruhat order, and corresponding Hasse diagrams. An important example is in the Coxeter groups.
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 ...