The Number of Conjugacy Classes of a Finite Group and its Sylow p-subgroups
Bret Benesh, Mathematics
The project will investigate the relationship between the number of conjugacy classes of a finite group and the number of conjugacy classes of its Sylow p-subgroups. The idea of this project comes from problem 14.74 of the Kourovka Notebook of Unsolved Problems in Group Theory (submitted by L. Pyber). The problem states: "Let k(H) denote the number of conjugacy classes of a group H, and G be a finite group with Sylow p-subgroups P1, ... , Pn. Prove or disprove: k(G) ≤ k(P1) ... k(Pn)". In this project, we will discuss my approach to this problem, some upper bounds of k(G), some lower bounds of k(Pi) and some families of groups for which this result holds.
Van, Cong Tuan Son, "The Number of Conjugacy Classes of a Finite Group and its Sylow p-subgroups" (2011). Honors Theses. 110.