Existence of finite indexed normal subgroup for a given finite indexed subgroup.
86336
For $G$ group and $H$ subgroup of finite index, prove that $N \subset H$ normal subgroup of $G$ of finite index exists
139611
Can a normal subgroup of a finite nonabelian group be nonabelian?
68955
Characterising subgroups of Prüfer $p$-groups.
110708
On proving that a finitely generated group has a finite number of subgroups with index $n$.
61373
A normal subgroup that is not a characteristic
57309
Show that the free group on $n$ generators is a finite index subgroup of $F_2$
121587
Let $H$ be a normal subgroup of index $n$ in a group $G$. Show that for all $g \in G, g^n \in H$
14255
Commutator subgroup and cyclic normal subgroup
158396
Finite $G$ with involutory automorphism $\alpha$ with no nontrivial fixed points. Proving properties of $\alpha$.
7108
Normal subgroup of prime index
61550
Existence of elements of order $p^k$ in a finite group
125244
Equivalent definitions of normal subgroups
106660
Every subgroup of the integers has finite index
94999
A subgroup with an index of 2 is normal to the group.
95534
If $|G|=p^nq$, then $G$ contains a unique normal subgroup of index $q$
64269
Finding the minimal $n$ such that a given finite group $G$ is a subgroup of $S_n$
78580
Is $SL_n(\mathbb{R})$ a normal subgroup in $GL_n(\mathbb{R})$?