Prove or disprove. Let $G$ be an abelian group. Let $N \triangleleft G$. Then, $ G\cong N \times G/N. $
71232
if $A$ is Abelian group , $B$ is subgroup of $A$ , Is $B \times A/B \cong A$?
99346
Prove a result on transitive group actions.
138406
If N and every subgroup of N is normal in G then G/N is abelian .
97565
Quotient group $G/H$ is abelian iff $[G,G] \subseteq H$
77221
Prove that $G$ has a subgroup isomorphic to $G/H$.
27905
Let $G$ be a group, where $(ab)^3=a^3b^3$ and $(ab)^5=a^5b^5$. Prove that $G $ is an abelian group.
49527
Prove that $a$ commutes with each of its conjugates in $G$ iff $a$ belongs to an abelian normal subgroup of $G$.
109611
If G is a group, prove that $(ab)^n =a^nb^n$ implies G is abelian.
79933
Let $G$ be a group and $c\in G$. Define $*$ on $G$ by: $x*y=xdy$ for all $x,y\in G$, where $d$ is the inverse of $c$. Prove $(G,*)$ is a group.
47179
Prove that $[N, G']=1$ when $N$ is a nilpotent subgroup of a group $G$?
78263
Prove $GL_n(\Bbb{R})/SL_n(\Bbb{R}) \cong \Bbb{R}^\times$
107687
How to prove that $U_{2^n}$ is isomorphic as group to $\mathbb Z_2 \times \mathbb Z_{2^{n-2}}$ for $n \ge 3$?
155219
Prove or Disprove: If every nontrivial subgroup of a group $G$ is cyclic, then $G$ is cyclic.
5741
Direct product of abelian groups
108425
Problem on abelian group
35188
Counterexample: $G \times K \cong H \times K \implies G \cong H$
82037
Suppose $G$ is a group generated by elements $x$ and $y$ where $xy^2 = y^3x$ and $yx^3 = x^2y$ What can you prove about $G$?