What can we say about invertible matrix $P$ under a condition that $A=PAP^{-1}$? Given an integral matrix $A$, we shall consider the set $$\{P:\text{invertible}|PAP^{-1}=A\}.$$
We can check that the set is a group under the matrix multiplication.
Can we say something algebraic for $P$? I am considering the concept of commutator to find some property for $P$. If you have another viewpoint or some comment, can you give it to me?
 A: Since you ask for something algebraic, the following is some more information that just commuting. 
Let $\mathbb{F}$ be a field. Denote by $C_A = \{B\in M_n(\mathbb{F}) | AB=BA\}$ the centralizer of a matrix $A$. See also a neat way by Amritanshu Prasad to find the vector space dimension of $C_A$ over $\mathbb{F}$. Denote by $G_A=\{P\in \mathrm{GL}_n(\mathbb{F})| AP=PA\}$. Then $G_A=\{P\in C_A| \det P \neq 0\}$. 
If we treat $\mathbb{F}^n$ as a $\mathbb{F}[x]$-module $M^A$ given by $x\cdot v=Av$, then by the structure theorem for finitely generated modules over PID, we may write 
$$
M^A\simeq \bigoplus\limits_p \mathop{\oplus}\limits_i \mathbb{F}[x]/(p^{\lambda_{p,i}}),
$$
where $p$ runs over irreducible factor of the characteristic polynomial of $A$ and $\lambda_{p,i}>0$ is the powers of $p$ appearing on the $p$-primary part of $M^A$. Then the centralizer of $A$ can be written as an $\mathbb{F}[x]$-module endomorphism algebra
$
\mathrm{End}_{\mathbb{F}[x]}(M^A). 
$
Then we need invertible matrices in $
\mathrm{End}_{\mathbb{F}[x]}(M^A). 
$ 
In case when $\mathbb{F}$ is a finite field, a method of obtaining such matrices $P\in G_A$ is outlined in Theorem 1.10.7 of Enumerative Combinatorics I. This method general and also applies when $\mathbb{F}$ is not a finite field. 
