Let $G$ be subgroup of $SL(n,Z)$ such that for any $g\in G$ there exists integer $m\geq1$ $g^m=1$.
Show that there exists $N\geq1$ such that for any $g\in G$ , $g^N=1$
I know $m$-th root of unity is the eigenvalue of elements, any element is diagonalizable matrix over complex number but I don't know how to use facts? any suggestion?