Are the conjugacy classes normal subgroups in $S_n$?
11043
A normal subgroup is the union of conjugacy classes.
118117
Conjugacy Class in a matrix group with columns of standard basis
96680
Generators of symmetric group $S_n$
91772
Proving that $A_n$ is the only proper nontrivial normal subgroup of $S_n$, $n\geq 5$
154878
Prove that $H$ is a normal subgroup in the finite group $G=\langle S\rangle $ iff $sHs^{-1}$ $\subseteq H$ for all $s\in S$.
33945
Showing that $|N \cap Z(G)| > 1$ for normal subgroups of p-groups
13523
Are normal subgroups transitive?
50762
If $S_n$ has cyclic subgroups of these orders or not?
57227
Maximal normal $\pi$-subgroups and torsion subgroups
11731
Normal subgroups of $S_4$
126930
Finding all group homomorphisms of $S_n\to\mathbb{C}^*$ and $A_n\to\mathbb{C}^*$
91874
Existence of finite indexed normal subgroup for a given finite indexed subgroup.
101079
Inner automorphisms form a normal subgroup of $\operatorname{Aut}(G)$
45580
Intersection of conjugate subgroups is normal
113013
Let $M$ and $N$ be two normal subgroups of $G$. Show that $M \cap N$ is also normal in $G$. Then show that $G/(M \cap N) \cong (G/M) \times (G/N)$
2254
Finite group with isomorphic normal subgroups and non-isomorphic quotients?
97865
On Conjugacy Classes of Alternating Group $A_n$