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A cycle graph illustrates the various cycles of a group and is particularly useful in visualizing the structure of small finite groups. A cycle graph for a cyclic group is simply a circular graph, where the group order is equal to the number of nodes. A single generator defines the group as a directional path on the graph, and the inverse ...
In group theory, a subfield of abstract algebra, a cycle graph of a group is an undirected graph that illustrates the various cycles of that group, given a set of generators for the group. Cycle graphs are particularly useful in visualizing the structure of small finite groups .
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 ...
Circulant graphs can be described in several equivalent ways: [2] The automorphism group of the graph includes a cyclic subgroup that acts transitively on the graph's vertices. In other words, the graph has an automorphism which is a cyclic permutation of its vertices. The graph has an adjacency matrix that is a circulant matrix.
A group that is generated by a single element is called cyclic. Every infinite cyclic group is isomorphic to the additive group of the integers Z. A locally cyclic group is a group in which every finitely generated subgroup is cyclic. The free group on a finite set is finitely generated by the elements of that set .
A presentation of a group by generators corresponds to a surjective homomorphism from the free group on generators to the group , defining a map from the Cayley tree to the Cayley graph of . Interpreting graphs topologically as one-dimensional simplicial complexes , the simply connected infinite tree is the universal cover of the Cayley graph ...
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Cycle graph, a connected, 2-regular graph; Cycle graph (algebra), a diagram representing the cycles determined by taking powers of group elements; Circulant graph, a graph with cyclic symmetry; Cycle (graph theory), a nontrivial path in some graph from a node to itself; Cyclic graph, a graph containing at least one graph cycle; Cyclic group, a ...