For any group G, let ρ be an
irreducible representation of the group algebra FG over a field F. Then by Schur’s
lemma, the center Δ of its commuting ring, is a field containing F. If ρ is
finite-dimensional over Δ, lhen it is called finite and if it is finite-dimensional over F
itself, then it is called strongly finite. In this paper, certain conditions are given for
finiteness of ρ. Also it is shown that for some types of groups, finiteness of ρ is related
to the existence of abelian subgroups of finite index in certain quotient of the group.
Conditions under which finiteness and strongly finiteness are equivalent, are given.
Finally, consequencesof ρ being faithful on G, or being faithful on FG, are
studied.