Some results on the structure of finite-dimensional cyclic modules for a semisimple Lie algebra are presented. Cyclic modules arise naturally in constructing symmetry adapted states of a system using projection. Projecting out states with definite symmetry from an arbitrary state ψ is related to the properties of the cyclic module generated by ψ. An important example of a cyclic module is the tensor product of two irreducible modules V(λ)⊗V(μ) which is cyclically generated by the vector vλ−⊗vμ+, where vλ−(resp., vμ+) is the minimal (resp., maximal) weight vector of V(λ) [resp., V(μ)]. For this particular case we determine the explicit form of the annihilator, in the universal enveloping algebra, of the cyclic vector vλ−⊗vμ+. It is hoped that this result may add new insight into the Clebsch–Gordan multiplicity problem. As an application of this result projection operators are constructed which project, from an arbitrary vector of weight λ, a maximal weight vector of weight λ.