Summer 2020 Projects

Mailing address

REU program
Dept. of Math Sciences
Kent State University
Math & CS Building
Summit Street, Kent OH 44242

Contact

Jenya Soprunova
reu [at] math.kent.edu
TEL: (330)672-9086
FAX: (330) 672-2209

This year we will recruit nine students to work on the two projects described below.

Ring Theory (Mikhail Chebotar)

This Summer my group will be working on some problems related to the area of Ring Theory. Possible topics may include functional identities; radical theory; interactions between Ring Theory and Linear Algebra.

For the previous projects of my REU groups please visit https://sites.google.com/a/kent.edu/mikhail-chebotar/reu-stude.

Prerequisites: Linear Algebra (Theory of Matrices), Abstract Algebra.

Graphs associated with groups (Mark Lewis)

This is a project focused on finite groups. We will consider several graphs associated with groups to better understand the group. Using graphs to study groups has a long history going back to Cayley. The graphs that we are interested in have been introduced more recently. These graphs take as their vertices a subset of the elements of the group, and then there is an edge between two vertices if the subgroup generated by the two elements has certain properties. Two graphs that we will particularly focus on are the commuting graph and the cyclic graph. The commuting graph takes all noncentral elements of the group as it vertices and it put an edge between those elements if the two elements generate an abelian group. The cyclic graph takes all nonidentity elements as its vertices and puts an edge between two elements if they generate a cyclic group. We will work on determining the properties of these graphs for p-groups, nilpotent groups, and solvable groups.

Prerequisites: At least one semester of undergraduate Abstract Algebra.

References:

Diane Imperatore and Mark L. Lewis, A condition in finite solvable groups related to cyclic subgroups, Bull. Aust. Math. Soc. 83 (2011), no. 2, 267-272.
Diane Imperatore, On a graph associated with a group, in: Proceedings of International Conference Ischia Group Theory (World Scientific Publishing, Singapore, 2008), pp. 100-115.
Christopher Parker, The commuting graph of a soluble group, Bull. Lond. Math. Soc. 45 (2013), no. 4, 839-848.
Michael Giudici and Chris Parker, There is no upper bound for the diameter of the commuting graph of a finite group, J. Combin. Theory Ser. A 120 (2013), no. 7, 1600-1603.
G. L. Morgan and C. W. Parker, The diameter of the commuting graph of a finite group with trivial centre. J. Algebra 393 (2013), 41-59.