IMPORTANT: If you are interested in this project, please indicate in your personal statement for VERSEIM that you are and whether it is your first, second, third or fourth choice.

Program Project Description:

A recently emerging perspective in the study of groups involves the use of graphs that encode particular algebraic properties of groups. Given a subgroup-closed class of finite groups C (for example, the class of abelian groups, nilpotent groups or solvable groups) and a finite group G, we define a graph whose vertices are the elements of G and where two vertices x and y are joined if the subgroup generated by x and y belongs to C. Properties of this graph can have dramatic consequences on the structure of G. Properties of interest include, the degree of vertices, connectedness, existence of Eulerian or Hamiltonian cycle and chromatic number.  When C is the class of solvable groups, the resulting graph is called the solubility graph.  Solubility graphs have been studied very broadly.  One important result is that if the degree of some non-identity element of the solubility graph of G is p-1, where p is a prime then G is an abelian simple group. Solubility graphs have also been characterized for an infinite class of finite groups called minimal simple groups.

However, there remain many questions about solubility graphs and graphs defined by other classes C. For example:

-Can we characterize these graphs for significant infinite classes of finite groups?

– What are the possible cardinalities of neighborhoods in these graphs?

-If x is a p-element, can we say more about the neighborhood of x?  How about, more specifically, if x is an element of order 2?  These elements often play a significant role in finding many interesting properties of these graphs.

During this project, we will explore these kinds of questions. We will be aided in our exploration through the use of the computer algebra system GAP. 

Applicants are expected to have taken a class in abstract algebra. Some experience with computer programming would be an asset but is not required.