Let $\mathsf{Grp}$ be the category of groups. A [bifunctor](https://en.wikipedia.org/wiki/Functor#Bifunctors_and_multifunctors) $A: \mathsf{Grp} \times \mathsf{Grp}\to \mathsf{Grp}$ is an *addition bifunctor* if:


* $A(C\_n,C\_m) \simeq C\_{n+m}$,
* $A(C\_0,G) \simeq A(G,C\_0) \simeq G$,


for every group $G$ and every $n,m$, with $C\_n$ the cyclic group of $n$ elements if $n>0$, and $C\_0 \simeq \mathbb{Z}$. 


**Question**: Is there an addition bifunctor for the category of groups?  

(or for the subcategory of countable groups) 


*Stronger question*: Is there an addition bifunctor providing a [monoidal structure](https://en.wikipedia.org/wiki/Monoidal_category)?


This post is inspired by [that one](https://mathoverflow.net/q/351019/34538) (without knowing whether there is an explicit link). 


Multiplicative analogous: [Existence of a multiplication bifunctor for the category of groups](https://mathoverflow.net/q/351502/34538).



Existence of an addition bifunctor for the category of groups
The answer is no. Notice that $A(-,C\_1)$ is a functor $F : \mathsf{Grp} \to \mathsf{Grp}$ with $F(C\_n) \cong C\_{n+1}$. But there is no such $F$. There is a split monomorphism $C\_1 \to C\_2$, hence $F$ would induce a split monomorphism $C\_2 \to C\_3$, contradiction.


