Parity Group is Only Group with 2 Elements

Theorem
Let $\struct {G, \circ}$ be a group with exactly $2$ elements.

Then $\struct {G, \circ}$ is isomorphic to the parity group, which can be exemplified $\struct {\Z_2, +_2}$.

That is, the additive group of integers modulo $2$.

Proof
We have that $2$ is a prime number.

Hence $\struct {\Z_2, +_2}$ is a prime group.

The result follows from Prime Groups of Same Order are Isomorphic.