74 min
- Source: Prime graph
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 These graphs were introduced in a 1981 paper by J. S. Williams, credited to unpublished work from 1975 by K. W. Gruenberg and O. Kegel.
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
p
{\displaystyle p}
and
q
{\displaystyle q}
for which there exists a group element with order
p
q
{\displaystyle pq}
.
Equivalently, there is an edge from
p
{\displaystyle p}
to
q
{\displaystyle q}
whenever the given group contains commuting elements of order
p
{\displaystyle p}
and of order
q
{\displaystyle q}
, or whenever the given group contains a cyclic group of order
p
q
{\displaystyle pq}
as one of its subgroups.
Properties
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.
Related graphs
Variations of prime graphs that replace the existence of a cyclic subgroup of order
p
q
{\displaystyle pq}
, 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.
References
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