Prove that if $G$ has no subgroups of index $2$, then every subgroup of index $3$ is normal
7108
Normal subgroup of prime index
106779
How to show if $H,K$ are normal subgroups in a group $G$ then $[H,K] \leq H \cap K$
54082
Group with exactly two subgroups of index 2
70373
Intersection of normal subgroups proof
8841
A $p$-group of order $p^n$ has a normal subgroup of order $p^k$ for each $0\le k \le n$
11731
Normal subgroups of $S_4$
95534
If $|G|=p^nq$, then $G$ contains a unique normal subgroup of index $q$
1996
$A_n$ is the only subgroup of $S_n$ of index $2$.
118337
Let $S$ be a normal p-subgroup of a finite group $G$. Prove that $S \subseteq P$ for every Sylow p-subgroup $P$ of $G$
89508
Existence of normal subgroups for a group of order $36$
58251
To show every sylow p-subgroup is normal in G
59561
Suppose that G is a group and N < G a normal subgroup. Prove G/N is simple
48273
If a maximal subgroup is normal, it has prime index
141265
Finding a normal and not normal subgroup of $S_3$
34902
If $H$ is a cyclic subgroup of $G$ and $H$ is normal in $G$, then every subgoup of $H$ is normal in $G$.
86336
For $G$ group and $H$ subgroup of finite index, prove that $N \subset H$ normal subgroup of $G$ of finite index exists
99114
Subgroups of index 3 are normal