Title of the course: Prime graphs of finite groups – an invitation to research
Instructor: Dr. Thomas Keller
Institution: Texas State University
Dates: 27 July – 2 August 2026
Prerequisites: Students should be familiar with group theory, group actions, Sylow theorems etc. Knowing some graph theory is helpful, but not needed for this course.
Level: Graduate students or advanced undergraduate students.
Abstract: In this course we will study prime graphs (also known as Gruenberg-Kegel graphs) of finite groups. For a group G, the prime graph Γ(G) is defined as follows: The vertices are the prime divisors of |G|, and there is an edge between primes p and q if and only if there is an element of order pq in G. Gruenberg and Kegel introduced this notion in the 1970s to study certain cohomological questions of group rings, but it has been studied for its own sake ever since. In 2015 the prime graphs of solvable groups were classified as follows. Theorem: An unlabeled simple graph is isomorphic to the prime graph of a solvable group if and only if its complement is 3-colorable and triangle-free. Since then, many extensions of this result have been obtained by “allowing in” one simple group at a time. In the course, we will introduce prime graphs, provide some examples and study their basic properties. We will also prove the above result and discuss some of the extensions, and then look at some very recent work on prime graphs that has been obtained in collaboration with undergraduate students and present open questions for future work.
Language: EN