Let $(G,\cdot)$ be a group. Define the law $\cdot$ on $\mathscr F(X,G)$ with the usual: $(f\cdot g )(x) = f(x)\cdot g(x)$.
I would like to say that $(\mathscr F(X,G), \cdot)$ is also a group without additionnal conditions (closure, associativity, neutral element and inverse seem easy to prove), but I've remembered that there is a trap (something to do with the commutativity).
Is my memory good? If not, do you see the result which has confused me?