For the subgroup $N$ of $S_3$, $N = \{(1),(123),(132)\}$, I calculate that $(13)N = \{(13),(123),(23)\}$ and $N(13) = \{(13),(23),(12)\}$. Shouldn't this show that $N$ is not a normal subgroup, as opposed to what's printed here?
2026-03-29 14:57:36.1774796256
Normal subgroup of $S_3$?
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Your computation of $(13)N$ is wrong. The $(123)$ should be $(12)$. One way to check this is that every permutation in $N$ is even, so the coset should consist only of odd permutations.