1. The problem asks to explain what a bilinear function is when defined from a Cartesian product to an abelian group $G$. 2. A bilinear function $f$ is a function defined on the Cartesian product of two groups or vector spaces, say $X \times Y$, mapping into an abelian group $G$: $$f: X \times Y \to G$$ 3. The key property of bilinearity means that $f$ is linear in each argument separately when the other is fixed. More precisely, for all $x,x' \in X$, $y,y' \in Y$, and for the group operation $+$ in $G$: - $f(x + x', y) = f(x,y) + f(x',y)$ - $f(x, y + y') = f(x,y) + f(x,y')$ 4. Since $G$ is abelian, the addition operation is commutative, which ensures the sums above are well-defined and independent of order. 5. Intuitively, bilinearity means the function respects the group structure in each variable independently, making it a natural generalization of linear maps to two variables. 6. This concept is fundamental in algebra, especially in tensor products, multilinear algebra, and representation theory. Final answer: A bilinear function from a Cartesian product $X \times Y$ to an abelian group $G$ is a function $f$ such that for fixed $y$, $f(\cdot,y)$ is a group homomorphism from $X$ to $G$, and for fixed $x$, $f(x,\cdot)$ is a group homomorphism from $Y$ to $G$.