Abstract: A vector field on a Lie Group is linear if its flow is a one parameter group of automorphisms. A linear system is obtained by adding left invariant controlled vector fields. The observability of such a system, whenever the output function is a Lie group morphism, was studied by Ayala and Hacibekiroglu. Within this framework it is shown that no observable system exists on semisimple groups, and necessary conditions for the existence of such a system on a general Lie group are given. The case where the output morphism is replaced by the projection on a homogeneous space is briefly discussed.