In this paper two theorems about harmonic maps are proved. The first result states the non-existence of non-trivial harmonic maps of minimum energy. This result is well known and there are even generalizations [see Y. L. Xin, Duke Math. J. 47, 609-613 (1980; Zbl 0513.58019); J. Eells and L. Lemaire, Selected topics in harmonic maps, Reg. Conf. Ser. Math. 50 (1983; Zbl 0515.58011); the reviewer, ”Maps of minimum energy from simply connected Lie groups”, to appear in Ann. Global Anal. Geom. 2 (1984)]. Moreover the computation of the index in the first assertion of Theorem 1 is incorrect, because the author assumes that the map has maximal rank. In Theorem 2 a reformulation of the condition of harmonicity for maps into Lie groups is given in terms of the equations satisfied by the pullback of the Maurer-Cartan form.