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.
Cayley Table
We can completely describe the parity group by showing its Cayley table:
- $\begin{array} {r|rr}
\struct {\set {1, -1} , \times} & 1 & -1\\ \hline 1 & 1 & -1 \\ -1 & -1 & 1 \\ \end{array}
\qquad
\begin{array} {r|rr} \struct {\Z_2, +_2} & \eqclass 0 2 & \eqclass 1 2 \\ \hline \eqclass 0 2 & \eqclass 0 2 & \eqclass 1 2 \\ \eqclass 1 2 & \eqclass 1 2 & \eqclass 0 2 \\ \end{array}
\qquad
\begin{array}{r|rr} + & \text{even} & \text{odd} \\ \hline \text{even} & \text{even} & \text{odd} \\ \text{odd} & \text{odd} & \text{even} \\ \end{array}$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 7$: Semigroups and Groups: Exercise $7.4$