Exercise 8, § 7

Let $n$ be an integer $\geqslant 1$. We write

where the product is taken over all prime numbers $l$ and the $r(l, n)$ are defined as in Exercises 6 and 7. Then $M(1) = 2, M(2) = 2^3 \cdot 3 = 24, M(3) = 2^4 \cdot 3 = 48,$ Deduce from Exercises 6 and 7 that the l.c.m. of the orders of the finite subgroups of $\mathbf{GL}(n, \mathbf{Q})$ (or $\mathbf{GL}(n, \mathbf{Z})$, which amounts to the same) is equal to $M(n)$. Consider the same question for $\mathbf{SL}(n, \mathbf{Q})$ with $M(n)$ replaced by $\frac{1}{2} M(n)$.