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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.
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 ...
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 =.
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 ...
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.
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 ...
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English: The number on the rim of the node circle indicates an order of an element the node represents. Node's colour marks conjugacy class of element with exception that elements of group's center have the same colour (light gray) despite every one of them being of its own conjugacy class of one element.