For $\mathsf{Grp}$ the category of groups, a bifunctor $M: \mathsf{Grp} \times \mathsf{Grp}\to \mathsf{Grp}$ is a *multiplication bifunctor* if:

- $M(C_n,C_m) \simeq C_{nm}$,
- $M(C_1,G) \simeq M(G,C_1) \simeq G$,

for every group $G$ and every $n,m>0$, with $C_n$ the cyclic group of $n$ elements.

**Question**: Is there a multiplication bifunctor for the category of groups?

(or for the subcategory of countable groups, or of finite groups)

*Stronger question*: Is there a multiplication bifunctor providing a monoidal structure?

This post is a multiplicative analogous of that additive one.