The Number of Conjugacy Classes of a Finite Group and its Sylow p-subgroups

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Mathematics | Physical Sciences and Mathematics


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.