Klein four-group

The Klein four-group, denoted $V_4$ (or $K_4$, or $\mathbb{Z}/2\times \mathbb{Z}/2$), is the smallest non-cyclic group:

$$V_4=\{e,a,b,c\}\text{with}{a}^2=b^2=c^2=e,ab=c,bc=a,ca=b.$$

Equivalently $V_4\cong C_2\times C_2$.

Properties

• Order: $4$.
• Abelian: every element commutes with every other.
• Every non-identity element has order 2.
• Three subgroups of order 2 (each generated by a non-identity element); five subgroups total ($\{e\}$, three order-2 subgroups, and $V_4$).

Realisations

• Symmetries of a non-square rectangle: identity, horizontal flip, vertical flip, 180° rotation.
• $V_4$ inside $S_4$: $\{e,(12)(34),(13)(24),(14)(23)\}$ - three double transpositions plus the identity. This is a Normal subgroup of $A_4$ (and also of $S_4$).
• Multiplicative group of ${\mathbb{F}}_4$ - wait, that’s $C_3$. The additive group $({\mathbb{F}}_4,+)$ IS $V_4$.
• Galois group of $\mathbb{Q}(\sqrt{2},\sqrt{3})/\mathbb{Q}$ - its three non-trivial elements correspond to the three sign-changes $\sqrt{2}\to \pm \sqrt{2}$, $\sqrt{3}\to \pm \sqrt{3}$.

Smallest Non-Cyclic Group

By Lagrange’s theorem, any group of order $\le 3$ is cyclic. Of the two groups of order 4 ($C_4$ and $V_4$), $V_4$ is the non-cyclic one - the smallest non-cyclic group.

Significance

• Inside $A_4$, $V_4$ is the unique non-trivial proper normal subgroup. This is why $A_4$ is not simple, in contrast to $A_n$ for $n\ge 5$.
• The first non-trivial example of a non-cyclic abelian group; an instance of the Fundamental Theorem of Finite Abelian Groups decomposition $V_4\cong C_2\times C_2$.