Isomorphic Groups Example I've been learning about isomorphisms (Two groups $(G, \cdot)$ and $(H, \circ)$ are isomorphic if there exists a one-to-one and onto map $\phi : G \rightarrow H$ such that the group operations is preserved; that is $\phi(a \cdot b) = \phi (a) \circ \phi (b)$ for all $a$ and $b$ in $G$.) but I was wondering if there's a group isomorphic to $\langle \lbrace 0 \rbrace, + \rangle$ other than itself? Could you give an example of a group that is isomorphic to  $\langle \lbrace 0 \rbrace, + \rangle$? 

There is not much choice. Since $G:=\langle \{ 0 \}, + \rangle$ has only one element, it is isomorphic to the group $H:=\langle  \{g\},+\rangle$ uniquely defined on any one element set $\{g\}$ with the trivial group operation $g+g=g$. In other words, there is a unique group with one element up to isomorphism.
