TITLE: Existence of a multiplication bifunctor for the category of groups
QUESTION [5 upvotes]: 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.

REPLY [8 votes]: 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.