Prime graph

This article is about graphs defined from finite groups. For other uses, see Glossary of graph theory § prime.

In the mathematics of graph theory and finite groups, a prime graph is an undirected graph defined from a group. These graphs were introduced in a 1981 paper by J. S. Williams, credited to unpublished work from 1975 by K. W. Gruenberg and O. Kegel.^[1]

Definition

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 ${\displaystyle p}$ and ${\displaystyle q}$ for which there exists a group element with order ${\displaystyle pq}$.^[1][2]

Equivalently, there is an edge from ${\displaystyle p}$ to ${\displaystyle q}$ whenever the given group contains commuting elements of order ${\displaystyle p}$ and of order ${\displaystyle q}$,^[1] or whenever the given group contains a cyclic group of order ${\displaystyle pq}$ as one of its subgroups.^[2]

Properties

Certain finite simple groups can be recognized by the degrees of the vertices in their prime graphs.^[3] The connected components of a prime graph have diameter at most five, and at most three for solvable groups.^[4] When a prime graph is a tree, it has at most eight vertices, and at most four for solvable groups.^[5]

Related graphs

Variations of prime graphs that replace the existence of a cyclic subgroup of order ${\displaystyle pq}$, in the definition for adjacency in a prime graph, by the existence of a subgroup of another type, have also been studied.^[2] 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.^[6]