Exercise 9, § 7

Suppose that $K$ is locally compact. Let $G = \mathbf{GL}(n, K)$. Let $G_1$ be the set of $g \in G$ which leaves stable a lattice of $K^n$ with respect to $A$. Let $G_2$ be the set of $g \in G$ which generate a relatively compact subgroup in $G$. Let $G_3$ be the set of $g \in G$ whose eigenvalues in an algebraic closure of $K$ are of

† For more details on this exercise and on the preceding one, see H. Minkowski, Gesamm. Abh., Leipzig-Berlin, Teubner, 1911 (Bd I, S. 212–218) and W. Burnside, Theory of groups of finite order (2nd ed.), Cambridge Univ. Press, 1911 (pp. 479–484).

absolute value 1. Then $G_1 = G_2 = G_3 = G_r$. (Use the argument of Exercise 5 (a).)