Every proper normal subgroup of a $p$-group contains non-trivial central elements
19720
A normal subgroup intersects the center of the $p$-group nontrivially
91874
Existence of finite indexed normal subgroup for a given finite indexed subgroup.
153921
Infinite nilpotent group, any normal subgroup intersects the center nontrivially
23533
All subgroups normal $\implies$ abelian group
29996
Prove that if a normal subgroup $H$ of $ G$ has index $n$, then $g^n \in H$ for all $g \in G$
33945
Showing that $|N \cap Z(G)| > 1$ for normal subgroups of p-groups
117744
$p$-Sylow in quotient groups
38095
The Commutator Subgroup is Normal.
113357
Groups of order $pqr$ and their normal subgroups
129830
A group with no proper non-trivial subgroups
14666
What are the non-trivial normal subgroups of $O(3)$?
101360
Why is a normal subgroup of $G_1\times G_2$ with trivial intersections with $G_1$ and $G_2$ is abelian?
116125
A group $G$ with $G'$ abelian and every abelian normal subgroup finite
14255
Commutator subgroup and cyclic normal subgroup
121481
Non-abelian finite $p$-group in which every element has prime order?
122335
Abelian $p$-group with unique subgroup of index $p$
43651
Normal subgroups and factor groups