Isomorphic Groups Example

Question

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

Answer 1

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.)

Answer 2

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.