Perlis-Walker theorem

Statement
Let $$G$$ and $$H$$ be finite Abelian groups. Then, the group algebra $$\mathbb{Q}(G)$$ is isomorphic to the group algebra $$\mathbb{Q}(H)$$ if and only if $$G$$ is isomorphic to $$H$$.

This is a step in the direction of solving the isomorphic group algebra problem, which asks for conditions under which algebra-isomorphic groups are isomorphic.