The Klein four-group is the smallest abelian group that is not cyclic. Recall that the Klein four-group is defined as

\begin{align*}
V = \langle a,b \ | \ a^2 = b^2 = e, (ab)^2 = e \rangle
\end{align*}

We will prove that the Klein four-group is abelian and not cyclic.

**Proof.** First we will prove that the Klein four-group is abelian, i.e., for all

a,b \in V we have

ab = ba. As Klein four-group is a group, it contains inverses. As

a^{-1} = a and

b^{-1} = b, we have that

\begin{align*}
(ab)^2 = abab = e & \iff ababb = b \\
& \iff aba = b \\
& \iff abaa = ba \\
& \iff ab = ba
\end{align*}

which proves that Klein four-group is abelian.
What is left to prove is that Klein four-group is not cyclic. We see that

V can’t be generated by

e,a or

b. As

ab,

(ab)^2 = e, Klein four-group can’t be generated by

ab. This proves that Klein four-group is not cyclic.