For $\mathsf{Grp}$ the category of groups, a [bifunctor](https://en.wikipedia.org/wiki/Functor#Bifunctors_and_multifunctors) $M: \mathsf{Grp} \times \mathsf{Grp}\to \mathsf{Grp}$ 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](https://en.wikipedia.org/wiki/Monoidal_category)?


This post is a multiplicative analogous of [that additive one](https://mathoverflow.net/q/351398/34538).



Existence of a multiplication bifunctor for the category of groups
No. $C\_1$ is a retract of $C\_2$, so $M(C\_2,C\_1)\simeq C\_2$ would have to be a retract of $M(C\_2,C\_2)\simeq C\_4$, which it isn't.


