Two $n\times n$ matrices $A$ and $B$ with entries in a field $\mathbb{F}$ are said to be similar if there exists $P\in GL_{n}(\mathbb{F})$ such that $B=PAP^{-1}$.
When $\mathbb{F}$ is an infinite field it is not hard to prove that the cardinality of the set containing all the distinct equivalence classes is the same of $\mathbb{F}$.
However, when the field $\mathbb{F}$ is finite, such as $\mathbb{Z}/p$, finding an expression for the number of distinct equivalence classes $S(n,p)$ seems to be a little bit harder.
Is there a nice way to derive a closed formula?