existence of a normal subgroup with certain properties
44942
A group $G$ with a subgroup $H$ of index $n$ has a normal subgroup $K\subset H$ whose index in $G$ divides $n!$
7973
Groups of order $pq$ have a proper normal subgroup
139611
Can a normal subgroup of a finite nonabelian group be nonabelian?
51747
Showing that the intersection of all subgroups of order $n$ is normal
62017
Distinct Sylow P subgroups
141265
Finding a normal and not normal subgroup of $S_3$
142413
Normal Subgroups of $SU(n)$
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 $G$
111049
Groups with given automorphism groups
139096
$|AxA|$ is a constant implies normal subgroup
72150
Union of the conjugates of a proper subgroup
23941
Normal Subgroups that Intersect Trivially
10602
Monotone Union of subgroups being subgroup
62603
Characterizations of the $p$-Prüfer group
2254
Finite group with isomorphic normal subgroups and non-isomorphic quotients?
78580
Is $SL_n(\mathbb{R})$ a normal subgroup in $GL_n(\mathbb{R})$?
8841
A $p$-group of order $p^n$ has a normal subgroup of order $p^k$ for each $0\le k \le n$