Abstract Algebra: Theory and Applications · Problem 9.4.50

The key is understanding how the group operation works on a direct product: two pairs multiply component-wise, so $(a_1, b_1)(a_2, b_2) = (a_1a_2, b_1b_2)$. Commutativity of the product depends entirely on whether each component commutes independently—there's no "mixing" between coordinates. This means the product is abelian exactly when you can swap elements in both slots simultaneously, which happens if and only if both groups already commute within themselves.

The key is understanding how the group operation works on a direct product: two pairs multiply component-wise, so $(a_1, b_1)(a_2, b_2) = (a_1a_2, b_1b_2)$. Commutativity of the product depends entirely on whether each component commutes independently—there's no "mixing" between coordinates. This means the product is abelian exactly when you can swap elements in both slots simultaneously, which happens if and only if both groups already commute within themselves.