Finite group in $GL(n,\mathbb{Q})$ is conjugate to finite group in $GL(n,\mathbb{Z})$?

Question

I can't solve this problem: Finite group in $GL(n,\mathbb{Q})$ is conjugate to finite group in $GL(n,\mathbb{Z})$. Could any one help me? Thanks a lot!

Answer 1

As Noam D. Elkies has said, every finite subgroup of $GL(n,\mathbb{Q})$ is conjugated to a subgroup of $GL(n,\mathbb{Z})$. The proof is well known, see for example in the book of J.P. Serre on Lie Algebras and Lie Groups, Appendix $3$, Theorem $1$. A crucial lemma is, that for a finite subgroup $H$ of $GL(n,\mathbb{Q})$ there exists a lattice $M$ which sends $H$ onto itself.