If $G$ is a group and $N$ is a nontrivial normal subgroup, can $G/N \cong G$?
27436
Does $G\cong G/H$ imply that $H$ is trivial?
11003
Proving that $G/N$ is an abelian group
131204
A non-trivial normal subgroup N of a finite group G or order $p^n$ is such that $N\cap Z(G)\neq \{ 1\}$
19458
Finite groups and Normal subgroups - $G$ is a finite group such that $|G|=n$. Let $p$ be the smallest prime such that $p|n$.
78580
Is $SL_n(\mathbb{R})$ a normal subgroup in $GL_n(\mathbb{R})$?
135701
An example of a group such that $G \cong G \times G$
54911
The preimage of a normal subgroup under a group homomorphism is normal
67272
Deduce that if $G$ is a finite $p$-group, the number of subgroups of $G$ that are not normal is divisible by $p$
34724
For $G$ a group and $H\unlhd G$, then $G$ is solvable iff $H$ and $G/H$ are solvable?
58033
$G\times H\cong G$ with $H$ non-trivial
7973
Groups of order $pq$ have a proper normal subgroup
113357
Groups of order $pqr$ and their normal subgroups
14666
What are the non-trivial normal subgroups of $O(3)$?
63162
Is there a characterization of groups with the property $\forall N\unlhd G,\:\exists H\leq G\text{ s.t. }H\cong G/N$?
115204
Show that SO(n) is a normal subgroup of O(n)
89508
Existence of normal subgroups for a group of order $36$
92493
If $G$ is non-abelian, then $Inn(G)$ is not a normal subgroup of the group of all bijective mappings $G \to G$