  We give an analog of Frobenius' theorem about the factorization of the group
determinant on the group algebra of finite abelian groups and we extend it into
dihedral groups and generalized quaternion groups. Furthermore, we describe the
group determinant of dihedral groups and generalized quaternion groups as a
circulant determinant of homogeneous polynomials.
  This analog on the group algebra is stronger than Frobenius's theorem and as
a corollary, we obtain a simple expression formula for inverse elements in the
group algebra. Furthermore, the commutators of irreducible factors of the
factorization of the group determinant on the group algebra corresponding to
degree one representations have interesting algebraic properties. From this
result, we know that degree one representations form natural pairing.
  At the current stage, the extension of Frobeinus' theorem is not represent as
a determinant. We expect to find a determinant expression similar to Frobenius'
theorem.
