Self-Maps of Low Rank Lie Groups at Odd Primes Let G be a simple, compact, simply-connected Lie group localized at an odd prime~p. We study the group of homotopy classes of self-maps $[G,G]$ when the rank of G is low and in certain cases describe the set of homotopy classes of multiplicative self-maps $H[G,G]$. The low rank condition gives G certain structural properties which make calculations accessible. Several examples and applications are given. Keywords:Lie group, self-map, H-mapCategories:55P45, 55Q05, 57T20