Let $\mathsf{Grp}$ be the category of groups. A bifunctor $A: \mathsf{Grp} \times \mathsf{Grp}\to \mathsf{Grp}$ 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?

This post is inspired by that one (without knowing whether there is an explicit link).

Multiplicative analogous: Existence of a multiplication bifunctor for the category of groups.