Let $A_1$, $A_2$ be groups and $\phi:A_1\to A_2$ surjective with $G_1\unlhd A_1$ and $\phi(G_1)=G_2$. Is $A_1/G_1\cong A_2/G_2$?
122581
Prove or disprove that $G_1/H_1 \cong G_2/H_2$
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 $G=HK$?
121449
An isomorphism from $GL_{2}(\mathbb{F_{2}})$ to $S_{3}$
144065
Proving $\phi(G)$ is cyclic if $\phi$ : $G \to H$ is group homomorphism and $G$ is cyclic
1026
Ring homomorphism with $\phi(1_R) \neq1_S$
7890
Prove the map $\Phi:G\to\operatorname{Aut}G$ with $g\mapsto (x\mapsto g^{-1}xg)$ is an homomorphism.
141683
$ |G_1 |$ and $|G_2 | $ are coprime. Show that $K = H_1 \times H_2$
11825
Let $H$ be a subgroup of $G$. Let $K = \{x \in G: xax^{-1} \in H \iff a \in H\}$. Prove that $K$ is a subgroup of $G$.
29545
$[(x-a_1)(x-a_2) \cdots (x-a_n)]^2 +1$ is irreducible over $\mathbb Q$
77221
Prove that $G$ has a subgroup isomorphic to $G/H$.
140334
Let $G$ be a group and $H$ be a subgroup of $G$ of index $2$ .Then show that if $a,b\notin H$ then $ab\in H$.
101543
Let $A$ be a group, where $a^2=1$, a belongs to $A$. Prove that this group is commutative.
66313
$\mathbb{Z}[i]/(a+bi) \cong \mathbb{Z}_{a^2+b^2}$ if $(a,b)=1$
87882
If $N \lhd G$ is abelian and $G/N$ is abelian then $G$ is abelian
154398
Show that H is a subgroup of G
132342
Suppose $H$ is a subgroup of $S_n$ but $H$ is not a subgroup of $A_n$. Prove that |$H$ ∩ $A_n$| = $\frac{1}{2}|H|$?
67199
Isomorphism $H_n(D^n, S^{n - 1};A)$ and $A$