If a group and its quotient group are isomorphic, the normal subgroup only has identity element?
27436
Does $G\cong G/H$ imply that $H$ is trivial?
116737
An element of a group $G$ is not conjugate to its inverse if $\lvert G\rvert$ is odd
58707
Prove Quotient Group Isomorphism
116168
Is $e^x$ the only isomorphism between the groups $(\mathbb{R},+)$ and $(\mathbb{R}_{> 0},*)$?
5996
Are $(\mathbb{R},+)$ and $(\mathbb{C},+)$ isomorphic as additive groups?
7973
Groups of order $pq$ have a proper normal subgroup
46260
Prove that a group generated by two elements of order $2$, $x$ and $y$, is isomorphic to $D_{2n}$, where $n = |xy|.$
123610
How to show that the commutator subgroup is a normal subgroup
123860
When are two direct products of groups isomorphic?
129753
Prove that a subgroup which contains half of all elements is a normal subgroup.
155054
$G$ be a group and $H$ be a normal subgroup of index $p$ ( a prime ) ; suppose $K$ is a subgroup of $G$ not contained in $H$ , then $G=HK$?
51359
Prove that every subgroup of an abelian group is a normal subgroup.
106779
How to show if $H,K$ are normal subgroups in a group $G$ then $[H,K] \leq H \cap K$
95534
If $|G|=p^nq$, then $G$ contains a unique normal subgroup of index $q$
99382
What is needed to make Euclidean spaces isomorphic as groups?
152603
Group isomorphic to its automorphism group
114522
Prove that identity is unique in a group