Answer to Question #23730 in Abstract Algebra for john.george.milnor

Question

For finite abelian groups G and H, show that RG ∼ RH as R-algebras iff |G| = |H| and |G/G2| = |H/H2|.

Answer 1

Since G is abelian, Wedderburn’s Theoremgives: RG ∼ R×· ··×R × C×· · ·×C.Suppose there are s factors ofR, and t factors of C, so that |G| = s + 2t. Thenumber s is the number of 1-dimensional real representations of G.This is the number of group homomorphisms from G to {±1} ⊆R*, so s = |G/G2|. Therefore, the isomorphism typeof RG (as an R-algebra) is uniquely determined by |G| and |G/G2|.The conclusion follows immediately from this.