Let $N$ be a cyclic normal subgroup of a group $G$ and $H$ is any subgroup of $N$. Prove $H$ is a normal subgroup of $G$
34902
If $H$ is a cyclic subgroup of $G$ and $H$ is normal in $G$, then every subgoup of $H$ is normal in $G$.
106277
Prove that $G$ contains a subgroup $H_i$ for each $i\in\Bbb Z$ with $\dots\subset H_{−2}\subset H_{−1}\subset H_0\subset H_1\subset\dots?$
131204
A non-trivial normal subgroup N of a finite group G or order $p^n$ is such that $N\cap Z(G)\neq \{ 1\}$
44152
If $G$ has only one subgroup of order $n$, then that Subgroup is Normal
51359
Prove that every subgroup of an abelian group is a normal subgroup.
14255
Commutator subgroup and cyclic normal subgroup
44934
Show that every normal subgroup can be written as an intersection of $\ker \chi$'s for $\chi \in \operatorname{Irr}(G).$
59561
Suppose that G is a group and N < G a normal subgroup. Prove G/N is simple
37971
Prove that the center of a group is a normal subgroup
65037
$H$ is a maximal normal subgroup of $G$ if and only if $G/H$ is simple.
123677
Normal subgroups of infinite symmetric group
102622
K is a subgroup of G. Show that K is not a normal subgroup
150042
Prove $x^{-1}Hx$ is a subgroup of $G$
23533
All subgroups normal $\implies$ abelian group
58251
To show every sylow p-subgroup is normal in G
45727
If $G$ contains a normal subgroup $H \cong \mathbb{Z_2}$ such that $G/H$ is infinite cyclic, then $G \cong \mathbb{Z} \times \mathbb{Z_2}$
113013
Let $M$ and $N$ be two normal subgroups of $G$. Show that $M \cap N$ is also normal in $G$. Then show that $G/(M \cap N) \cong (G/M) \times (G/N)$