  Motivated by an eigenvalue-eigenfunction problem posed in IR^n x {\Omega},
where {\Omega} is a probability space, we are concerned in this paper with the
Sobolev space on groups. Hence it is established an equivalence between locally
compact Abelian groups and the space of solutions to the associated variational
problem. Then, we study some conditions which characterize in a precisely
manner the Rellich-Kondrachov Theorem, the principal ingredient to solve the
variational problem.
