Prime graph

The prime graph of a group has a vertex for each prime number that divides the order (number of elements) of the given group, and an edge connecting each pair of prime numbers and for which there exists a group element with order .

Equivalently, there is an edge from to whenever the given group contains commuting elements of order and of order , or whenever the given group contains a cyclic group of order as one of its subgroups.

Certain finite simple groups can be recognized by the degrees of the vertices in their prime graphs. The connected components of a prime graph have diameter at most five, and at most three for solvable groups. When a prime graph is a tree, it has at most eight vertices, and at most four for solvable groups.

Variations of prime graphs that replace the existence of a cyclic subgroup of order , in the definition for adjacency in a prime graph, by the existence of a subgroup of another type, have also been studied. Similar results have also been obtained from a related family of graphs, obtained from a finite group through the degrees of its characters rather than through the orders of its elements.