zero group as zero object
1. Proposition
The zero group is the zero object of the category Group
2. Proof
2.1. terminal object
The only map is
\begin{align*} \varphi: G \rightarrow 0 \\ g \mapsto 0 \end{align*}2.2. initial object
Let \(G\) be a group, then the only welldefined grouphomomorphism is
\begin{align*} \varphi: 0 \rightarrow& G \\ 0 \mapsto e \end{align*}