Show that if $f : G → H$ is a homomorphism, $ H$ is abelian and $ N $ is a subgroup of $G$ containing $Ker f$, then $N$ is normal in $G$
109236
Subgroups containing kernel of group morphism to an abelian group are normal.
28722
Prove that $H$ is a abelian subgroup of odd order
49527
Prove that $a$ commutes with each of its conjugates in $G$ iff $a$ belongs to an abelian normal subgroup of $G$.
97605
Let $f$ be a surjective homomorphism. Prove that $\ker(f)$ is a maximal ideal
44152
If $G$ has only one subgroup of order $n$, then that Subgroup is Normal
113013
Let $M$ and $N$ be two normal subgroups of $G$. Show that $M \cap N$ is also normal in $G$. Then show that $G/(M \cap N) \cong (G/M) \times (G/N)$
46342
Prove that $G$ is abelian iff $\varphi(g) = g^2$ is a homomorphism
106809
$f: G→G$ defined by $f(x) =x^2$ is a homomorphism if and only if $G$ is abelian.
14255
Commutator subgroup and cyclic normal subgroup
34902
If $H$ is a cyclic subgroup of $G$ and $H$ is normal in $G$, then every subgoup of $H$ is normal in $G$.
101079
Inner automorphisms form a normal subgroup of $\operatorname{Aut}(G)$
45727
If $G$ contains a normal subgroup $H \cong \mathbb{Z_2}$ such that $G/H$ is infinite cyclic, then $G \cong \mathbb{Z} \times \mathbb{Z_2}$
11003
Proving that $G/N$ is an abelian group
111886
Let $(G,•)$ be a group in which $(a•b)^3 = a^3•b^3$. Prove that $H=\{x^3 : x \in G\}$ is a normal subgroup of $G$.
23533
All subgroups normal $\implies$ abelian group
80186
How to show that $K\leq S_4$ is a normal subgroup?
27670
Group homomorphisms $\mathbb{Q}\to\mathbb{R}$