Group categories with only one object with a defined product

Question

Do you know how to deal with this kind of problem?

Let $G$ be a group and $\mathcal {G} $ be the category with one object $G$ and $\mbox {Mor}( {G} ; {G} ) = {G} $. Find all groups $G$ such that the product in $\mathcal {G}$ is defined.

Answer 1

I'm going to assume that the question should look like this:

Let $G$ be a group and $\mathcal{G}$ be the category with one object $\star$ with $\operatorname{Mor}(\star, \star ) = G$ (as monoids). Find all groups $G$ such that the cartesian product is defined in $\mathcal {G}$.

In this case, if $\star\times\star$ exists, then we must have $\star\times\star = \star$, and $\operatorname{Mor}(\star, \star )=\operatorname{Mor}(\star\times\star, \star ) = \operatorname{Mor}(\star, \star )\times \operatorname{Mor}(\star, \star )$, where the identification is via two projection maps $\pi_1,\pi_2\in G$.

In other words, the map $g\mapsto (\pi_1 g, \pi_2 g)$ must be an isomorphism $G\to G\times G$. But then, for any $(a,b)\in G\times G$, we have $ab^{-1} = \pi_1 \pi_2^{-1}$, which can only occur if $G$ is the trivial group.

I could be wrong about how to interpret the question, but if so, a lot of clarification is in order.