Prove that there is no group $G$ such that it has non-trivial expansion simultaneous in direct product and free product, i.e. $G = A \times B = C * D$.
Check my attempt please.
Let's take an element $G \ni g = c * d$ such that $1 \neq c \in C, 1 \neq d \in D$ and look at it from the angle of free product. It commutes only with it's powers, so $C(g) \simeq \mathbb{Z}$.
Let's look at this group from the angle of direct product. Let's take the same element $g$ now $g = (a, b)$ for some $a \in A, b \in B$. Then centralizator $C(g) = C(a) \times C(b)$.
Now we have $\mathbb{Z} = C(a) \times C(b)$ and both $C(a), C(b)$ are non-trivial. But $\mathbb{Z}$ can not be decomposed in a direct product. Contradiction.