= Tetrahedral quotient onto the cyclic group of order three
{title2=$A_4/V_4\cong C_3$}
The identity and three half-turns form a normal <Klein four-group> $V_4$ inside the proper rotational <tetrahedral symmetry> group $A_4$. Conjugation preserves double-transposition cycle type, establishing normality. The <quotient group> has order three and is therefore cyclic. If $r$ is any 3-cycle, every element is uniquely $r^jk$ with $j\in\{0,1,2\}$ and $k\in V_4$. The formula $r^jk\mapsto e^{2\pi ij/3}$ defines a surjective <group homomorphism> with <kernel> $V_4$. Normality allows the factors from $V_4$ to be moved past powers of $r$, making exponents add modulo three and proving the homomorphism property directly.
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