Klein four-group is abelian and not cyclic

The Mathematician — Fri, 11 Nov 2022 13:00:53 +0000

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

b

. As

ab

(ab)^2 = e

, Klein four-group can’t be generated by

ab

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

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Klein four-group is abelian and not cyclic