Assume $n \ge 5$ and let $ H \triangleleft S_n$ be a normal subgroup. If $H$ contains at least one $3$-cycle, prove $H = S_n$ or $H = A_n$. I have no idea how to do this? I think i need to use the fact every element in $A_n$ is a product of $3$-cycles.
2026-03-28 10:35:57.1774694157
Question about 3-cycles?
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Yes, you are right. Just note that any two $3$-cycles are conjugate in $S_n$, so if the normal subgroup $H$ contains one $3$-cycle, it contains all of them.
More generally, two elements of $S_{n}$ are conjugate iff they have the same cycle structure.