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  $\langle \lbrace 0 \rbrace, + \rangle$? 

The group $\langle \{1\}, \times\rangle$ is isomorphic to $\langle \{0\}, +\rangle$ because they are both groups containing only the unit of the operation. The isomorphism $0\leftrightarrow 1$ preserves the unit, and every possible application of the group operation $0+0 = 0 \iff 1\times 1 = 1$.
Other examples include $\langle \{\varnothing\}, \cup\rangle$ and $\langle \{\mathbb{N}\}, \cap\rangle$. 
(Normally, these operations might not qualify as group operations because they're not invertible, but for a singleton group like this, all we really need is the unit property.)
