Every subgroup of finite index contained in an infinite group $G$ contains a normal subgroup of $G$.
91874
Existence of finite indexed normal subgroup for a given finite indexed subgroup.
37971
Prove that the center of a group is a normal subgroup
151619
A finite abelian group containing a non-trivial subgroup which lies in every non-trivial subgroup is cyclic
145511
Suppose $H$ is the only subgroup of order $o(H)$ in the finite group $G$. Prove that $H$ is a normal subgroup of $G$.
23533
All subgroups normal $\implies$ abelian group
131204
A non-trivial normal subgroup N of a finite group G or order $p^n$ is such that $N\cap Z(G)\neq \{ 1\}$
33945
Showing that $|N \cap Z(G)| > 1$ for normal subgroups of p-groups
109517
A subgroup such that every left coset is contained in a right coset.
15630
Normal subgroup of prime order is characteristic
129753
Prove that a subgroup which contains half of all elements is a normal subgroup.
125244
Equivalent definitions of normal subgroups
94999
A subgroup with an index of 2 is normal to the group.
58205
$ N $ normal in a finite group $ G $, $ |N| = 5 $ and $ |G| $ odd. Why is $ N \subseteq Z(G) $?
7108
Normal subgroup of prime index
142228
Proof that a subset closed under group operation of a finite group is a subgroup
91772
Proving that $A_n$ is the only proper nontrivial normal subgroup of $S_n$, $n\geq 5$
97928
Normal subgroup of a normal subgroup