Exercise 7, § 7

In this exercise, let $v_2(a)$ denote the 2-adic valuation of an integer $a$.

(a) Let $\Gamma$ be a finite subgroup of $\mathbf{GL}(n, \mathbf{Q})$. Show that there exists a positive definite quadratic form with coefficients in $\mathbf{Z}$, which is invariant under $\Gamma$. Deduce (by the same argument as in Exercises 4 and 5) that, for all sufficiently large $p$, $\Gamma$ is isomorphic to a subgroup of an orthogonal group $\mathbf{O}(n)$ over the field $\mathbf{F}_p$ (resp. a subgroup of $\mathbf{SO}(n)$ if $\Gamma$ is contained in $\mathbf{SL}(n, \mathbf{Q})$).

(b) Suppose that $\Gamma$ is contained in $\mathbf{SL}(n, \mathbf{Q})$ and let $2^e$ denote the greatest power of 2 dividing the order of $\Gamma$. Show that, if $n$ is odd, $2^e$ divides all the integers

for sufficiently large prime $p$. (Use (a) and Exercise 13 of Algebra, Chapter IX, § 6.)

Show that, if $n$ is even and $p$ is a sufficiently large prime, $2^e$ divides the l.c.m. $b_n(p)$ of the integers

(Same method as for $n$ odd.) (c) Under the hypotheses of (b), we write Let $A$ be an integer $\geqslant 3$. Show that $2^{r(2, n)-1}$ is the greatest power of 2 which divides all the $b_n(p)$ for $p \geqslant A$. (Same method† as in Exercise 6; use the existence of a prime number $p \geqslant A$ such that $p \equiv 5 \pmod{8}$.) Deduce the inequality $e \leqslant r(2, n) - 1$. (d) Conversely, let $C_n$ be the subgroup of $\mathbf{GL}(n, \mathbf{Z})$ generated by the permutation matrices and the diagonal matrices with coefficients $\pm 1$. The order of $C_n$ is $2^{nn!}$ and $v_2(2^{nn!}) = r(2, n)$. If $\Gamma_{2, n}$ denotes the intersection of a Sylow 2-group of $C_n$ with $\mathbf{SL}(n, \mathbf{Z})$, deduce that the order of $\Gamma_{2, n}$ is $2^{r(2, n)-1}$.