LetG andH be finite abelian groups and letF be an arbitrary field. One fundamental problem is that of determining necessary and sufficient conditions for the isomorphism
of the group algebrasFG andFH. No solution has appeared in the literature. Nevertheless by combining the results of Berman, Perlis and Walker, Cohen, and
Deskins and providing connecting arguments a complete solution can be obtained. It is the purpose of this note to present
such a solution.