# Matrix similar to a permutation matrix over $Z$

Suppose we have $$m\times m$$ matrices $$B,C$$ with integer entries such that: $$A^{-1}BA=C.$$ for some matrix $$A$$ with integer entries. ($$A^{-1}$$ may not have integer entries). Now, suppose we also know that $$C$$ is a permutation matrix and that $$det(A)$$ is co-prime to the order of $$C$$ (as a permutation). Then, does there necessarily exist $$A'\in GL_m(\mathbb{Z})$$, so that: $$A'^{-1}BA'=C?$$ This question has its roots in the representation theory of finite groups. I know that the above isn't true if we remove the condition that $$C$$ is a permutation matrix. For a counterexample, see this. Does having this extra condition change the answer?

• I guess you also want $A\ne 0$. – Saucy O'Path Oct 28 '18 at 23:12
• @SaucyO'Path Yup! Edited. – MathManiac Oct 28 '18 at 23:16