Curve selection Lemma for semianalytic sets and conjugacy classes of finite order in Lie groups

Abstract

Using a strong version of the Curve Selection Lemma for real semianalytic sets, we prove that for an arbitrary connected Lie group G, each connected component of the set E n (G) of all elements of order n in G is a conjugacy class in G. In particular, all conjugacy classes of finite order in G are closed. Some properties of connected components of E n (G) are also given.