Pure Core Sets of $n\times n$ Matrices over Finite Fields

This paper studies the structure of core sets under different similarity classes. We investigate the influence of factors of the minimal polynomial with different degrees on the structure of core sets. When $F$ is a finite field of prime order, we study the upper bound on the size of a non-core set in a similarity class in $M_n(F)$. We prove that as $\mathrm{\mid }F\mathrm{\mid }$ increases, the proportion of pure core sets among subsets of $M_n(F)$ tends to $1$.