# Schur multiplier of direct product

Let $$A$$ and $$B$$ be arbitrary finite groups with $$(|A|,|B|)=1$$. Let $$M(G)$$ be the Schur multiplier of the group $$G$$. Problem 5A.8.(b) in Isaacs' Finite Group Theory asks us to show that $$M(A \times B) \cong M(A) \times M(B)$$.

We have that $$|M(A \times B)| \ge |M(A)| |M(B)|$$ by (a). I anticipate an argument using Schur-Zassenhaus, but I am having trouble applying the coprime condition to the structure of the extension.