On Everywhere Dense Imbedding of Free Groups in Lie Groups

In this note it will be proved that some kinds of Lie groups (including semi-simple Lie groups) have an everywhere dense subgroup which is algebraically isomorphic to the free group generated by two elements (Theorem 8), In § 1 characterizations of Lie groups which are approximated by discrete subgroups’ are given. This section is closely connected with a part of the results of Malcev [4] and Matsushima [5], and some theorems are slight modifications of them (Theorems 2 and 3).