Project Combinatorics of Cyclic and Abelian Group Actions Program Purdue Experimental Mathematics Lab Term Spring 2026 Status Accepted Research Area Mathematics Description One can recover the orbit decomposition of a cyclic group action on a finite set from knowledge of the number of orbits associated with each subgroup. This property fails for the Klein four group, so it seems reasonable that finite cyclic groups might be the only groups enjoying this "orbit decomposition reconstruction" property. We will explore this idea and possibly search also for generalizations that do hold for larger classes of Abelian groups. Supervisor Thomas J Sinclair Mentor Type of work required Course-based, vertically-integrated research projects in mathematics. Each project will consist of a small research team consisting of typically 2-4 undergraduates, a graduate mentor, and a faculty mentor. The graduate mentor and undergraduates will meet on a weekly basis, with full team meetings every few weeks as determined by the faculty mentor. To apply include a brief (one page or less) statement explaining your interest in mathematics research. Additionally, list all mathematics courses you have taken with your grade in each one, as well as any other coursework or qualifications that you feel are pertinent. Undergraduates who have been accepted into a project must sign up for the 3-credit "Purdue Experimental Math Lab" course (currently listed under MA 490) and must pledge that they are able to dedicate 10 hours of effort per week to the project. Websites https://www.math.purdue.edu/pxml/join-pxml.html Qualifications Completion of courses in Linear Algebra and Abstract Algebra Credit Hours 3 Weekly Hours 10 (estimated)