Cyclic group of order four and Klein four-group ($C_4\ncong C_2\times C_2$)

The finite cyclic group $C_4=\mathbb{Z}/4\mathbb{Z}$ and the Klein four-group $V_4=C_2\times C_2$ are both abelian groups of order four. In $C_4$, composition is addition modulo four. In $V_4$, label the elements $00,01,10,11$ and add each coordinate modulo two. For example, $01+10=11$ and $11+11=00$.

The order of a group element is preserved by a group isomorphism: an equation $g^m=e$ holds exactly when its image satisfies the same equation. In $C_4$, both $1$ and $3$ have order four. In $V_4$, each nonidentity element has order two. Hence these groups are not isomorphic, despite having the same order of a finite group. No relabelling can repair this discrepancy.

For a geometric model of $V_4$, take the symmetries of a non-square rectangle: the identity element, reflection in its horizontal axis, reflection in its vertical axis, and rotation through $180^{\circ }$. The two reflections commute, and their composite is the half-turn. Every nonidentity motion is its own inverse element. This makes the two binary switches concrete: choosing whether to apply each reflection specifies all four motions. The case of abelian groups of prime-square order generalizes this distinction from $p=2$ to any prime $p$.