A group is simple if an only if its homomorphic images are the trivial group and $G$ itself (up to isomorphism)
8066
Sort-of-simple non-Hopfian groups
92899
$G$ is non abelian simple group of order $<100$ then $G\cong A_5$
55393
Existence of a group isomorphism between $(\mathbb K,+)$ and $(\mathbb K^\times,\cdot)$
90616
When is mapping $g$ to $g^{-1}$ a group homomorphism?
23279
Group Theory Isomorphism
33455
Question on group homomorphisms involving the standard Z-basis
112993
Is every quotient of a finite abelian group $G$ isomorphic to some subgroup of $G$?
104112
Number of finite simple groups of given order is at most $2$ - is a classification-free proof possible?
24181
Showing that $G$ is a group under an alternative operation.
3213
There exists only two groups of order $p^2$ up to isomorphism.
23579
If $G$ is isomorphic to $H$, then ${\rm Aut}(G)$ is isomorphic to ${\rm Aut}(H)$
114320
Showing that if the center of a group is trivial, then the center of its automorphism group is trivial
128300
Prove that a group with exactly two proper nontrivial subgroups is isomorphic to $\mathbb{Z}_{pq}$ or $\mathbb{Z}_{p^3}$.
146969
Proving that every group of order $4$ is isomorphic to $\Bbb Z_4$ or $\Bbb Z^*_8$
143451
Homomorphism from $(\Bbb Q,+)$ to a finite group
140962
Counterexamples for $G \times H$ isomorphic to $G$ implies $H$ trivial.
25421
Simple or not simple Groups