Let $G$ be Abelian. Then any subgroup of $G$ is normal. Does the converse hold?
58694
Which non-abelian finite groups have the property that every subgroup is normal?
14666
What are the non-trivial normal subgroups of $O(3)$?
104505
Is every normal subgroup the kernel of some endomorphism?
58251
To show every sylow p-subgroup is normal in G
16804
Prove whether or not a subgroup is normal in the dihedral group $D_4$
112993
Is every quotient of a finite abelian group $G$ isomorphic to some subgroup of $G$?
55976
Are groups satisfying $g=g^{-1}$ for all $g \in G$ abelian?
22477
If H is a subgroup of a group G and K is a normal subgroup of G the show that K is a normal subgroup of HK.
34902
If $H$ is a cyclic subgroup of $G$ and $H$ is normal in $G$, then every subgoup of $H$ is normal in $G$.
103423
Normal subgroup where G has order prime
150585
Group with $a=a^{-1}$ for all $a\in G$ is abelian
27703
an example of a non-abelian group $G$ containing a proper normal subgroup $N$ such that $G/N$ is abelian.
87882
If $N \lhd G$ is abelian and $G/N$ is abelian then $G$ is abelian
116125
A group $G$ with $G'$ abelian and every abelian normal subgroup finite
1364
Let $G$ be a group in which $a^2=e$ for all elements of $a$ of $G$. Show that $G$ is Abelian.
57464
If $G$ is a group show that if $(a \cdot b)^2 = a^2 \cdot b^2$ then $G$ must be abelian.
57227
Maximal normal $\pi$-subgroups and torsion subgroups