= Cyclic group of order four and Klein four-group
{title2=$C_4\not\cong C_2\times C_2$}
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 $11+11=00$.
\Image[/cyclic-four-versus-klein-four.png]
{title=Two abelian groups of order four have different addition tables}
{description=Each cell is the row element plus the column element. Diagonal cells record adding an element to itself. In $C_4$, only $0$ and $2$ return to zero after two additions. In the <Klein four-group>, every diagonal cell is the <identity element> $00$. The symmetric tables show <commutativity>, while the different doubling behavior distinguishes the <groups>.}
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 \b[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> $p$.
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