Given a subgroup $H $, find a normal subgroup of $G$ with index less than or equal to $n!$
44942
A group $G$ with a subgroup $H$ of index $n$ has a normal subgroup $K\subset H$ whose index in $G$ divides $n!$
125498
Check whether subgroup is normal in $ S_4$
7973
Groups of order $pq$ have a proper normal subgroup
55350
H is a subgroup of G and G' is a subgroup of H. Prove H is normal in G.
2174
Proving that HK is a subgroup when K is normal
122266
equivalent definition of normal subgroup
64408
Subgroup of $\mathbb{Q}$ with finite index
98542
Prove that $H$ is a subgroup of an abelian group $G$
43651
Normal subgroups and factor groups
98739
If $H$ is a subgroup of $G$ and $K$ is a subgroup of $H$, then $$|G:K|=|G:H||H:K|$$
49666
Generalization of index 2 subgroups are normal
7108
Normal subgroup of prime index
133001
$G$ simple group and there exists a subgroup of index $n$. Show that: $|G|$ divides $n!/2$
141265
Finding a normal and not normal subgroup of $S_3$
95534
If $|G|=p^nq$, then $G$ contains a unique normal subgroup of index $q$
151029
The number of non normal subgroups of a group cannot be $1$?
132342
Suppose $H$ is a subgroup of $S_n$ but $H$ is not a subgroup of $A_n$. Prove that |$H$ ∩ $A_n$| = $\frac{1}{2}|H|$?