If G contains a normal subgroup $N \cong \mathbb{Z}_2$ and $G/N \cong \mathbb{Z}$, then $G\cong \mathbb{Z}\times \mathbb{Z_2}$.
45727
If $G$ contains a normal subgroup $H \cong \mathbb{Z_2}$ such that $G/H$ is infinite cyclic, then $G \cong \mathbb{Z} \times \mathbb{Z_2}$
44152
If $G$ has only one subgroup of order $n$, then that Subgroup is Normal
34902
If $H$ is a cyclic subgroup of $G$ and $H$ is normal in $G$, then every subgoup of $H$ is normal in $G$.
67772
How to find $k$ such that $(\mathbb{Z} \times \mathbb{Z})/ \langle (m,n)\rangle \cong \mathbb{Z} \times \mathbb{Z}_k$
103423
Normal subgroup where G has order prime
4883
Let $H$ be a subgroup of a group $G$ such that $x^2 \in H$ , $\forall x\in G$ . Prove that $H$ is a normal subgroup of $G$
140471
Is $ G \cong G/N \times N$?
12455
$|G|>2$ implies $G$ has non trivial automorphism
98765
Showing that $G/(H\cap K)\cong (G/H)\times (G/K)$
19458
Finite groups and Normal subgroups - $G$ is a finite group such that $|G|=n$. Let $p$ be the smallest prime such that $p|n$.
87882
If $N \lhd G$ is abelian and $G/N$ is abelian then $G$ is abelian
27703
an example of a non-abelian group $G$ containing a proper normal subgroup $N$ such that $G/N$ is abelian.
154878
Prove that $H$ is a normal subgroup in the finite group $G=\langle S\rangle $ iff $sHs^{-1}$ $\subseteq H$ for all $s\in S$.
121587
Let $H$ be a normal subgroup of index $n$ in a group $G$. Show that for all $g \in G, g^n \in H$
43651
Normal subgroups and factor groups
135701
An example of a group such that $G \cong G \times G$
15276
$G \times H \cong G \times K$ , then $ K \cong H$