H a subgroup of index n, then G has a normal subgroup K with [G : K] ≤ n!.
44942
A group $G$ with a subgroup $H$ of index $n$ has a normal subgroup $K\subset H$ whose index in $G$ divides $n!$
33945
Showing that $|N \cap Z(G)| > 1$ for normal subgroups of p-groups
50706
$|G|=p(p+1)$ for $p$ prime, then $G$ has a normal subgroup of order $p$ or $p+1$
4883
Let $H$ be a subgroup of a group $G$ such that $x^2 \in H$ , $\forall x\in G$ . Prove that $H$ is a normal subgroup of $G$
34902
If $H$ is a cyclic subgroup of $G$ and $H$ is normal in $G$, then every subgoup of $H$ is normal in $G$.
111375
Normality of subgroups and Sylow groups under homomorphisms...
131204
A non-trivial normal subgroup N of a finite group G or order $p^n$ is such that $N\cap Z(G)\neq \{ 1\}$
22444
Normal subgroups of dihedral groups
44152
If $G$ has only one subgroup of order $n$, then that Subgroup is Normal
142413
Normal Subgroups of $SU(n)$
145511
Suppose $H$ is the only subgroup of order $o(H)$ in the finite group $G$. Prove that $H$ is a normal subgroup of $G$.
65972
Show that a group of order $108$ has a normal subgroup of order $9$ or $27$.
77221
Prove that $G$ has a subgroup isomorphic to $G/H$.
86336
For $G$ group and $H$ subgroup of finite index, prove that $N \subset H$ normal subgroup of $G$ of finite index exists
106779
How to show if $H,K$ are normal subgroups in a group $G$ then $[H,K] \leq H \cap K$
6134
Some Subgroup of Dihedral Group is Normal
41237
$H$ is normal subgroup of $G$ iff $gHg^{-1}=H$ for every $g$ of $G$