How many homomorphisms are there from $\mathbb Z_3$ to $S_4$?
I got this on a quiz today and calculated 9 but I'm not certain I was right. I sent 1 to id and to the 8 3-cycles... did I do wrong?
2026-04-11 12:15:21.1775909721
How many homomorphisms are there from $\mathbb Z_3$ to $S_4$?
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An homomorphism $f : \mathbb{Z}_{3} \to S_{4}$ is uniquely determined by $f(1)$, which must be an element of order that divides $3$, then ord$(f(1)) = 1$ or ord$(f(1)) = 3$.
In $S_{4}$ the only elements of order $3$ are the $3$-cycles, so your solution seems correct.