Alternate proof for : If $Z$ is the center of the group $G$ and $G/Z$ is cyclic, prove that $G$ is abelian.
23394
If $G/Z(G)$ is cyclic, then $G$ is abelian
6153
Suppose that $G$ is a group with the property that for every choice of elements in $G$, $axb=cxd$ implies $ab=cd$. Prove that $G$ is Abelian.
3756
If $G$ is abelian, then the set of all $g \in G$ such that $g = g^{-1}$ is a subgroup of $G$
74978
How the center of a non-cyclic free Group is trivial?
40287
Prove: if $a,b\in G$ commute with probability $>5/8$, then $G$ is abelian
4838
For $N\unlhd G$ , with $C_G(N)\subset N$ we have $G/N$ is abelian
14003
$(\mathbb{Z}/2^n \mathbb{Z})^*$ is not cyclic Group for $n\geq 3$
144065
Proving $\phi(G)$ is cyclic if $\phi$ : $G \to H$ is group homomorphism and $G$ is cyclic
125559
In a group $G$, if for all $a,b,c\in G$, $ab=ca\Rightarrow b=c$, then $G$ is abelian
7015
Prove that if $(ab)^i = a^ib^i \forall a,b\in G$ for three consecutive integers $i$ then G is abelian
51359
Prove that every subgroup of an abelian group is a normal subgroup.
127978
Another point of view that $\mathbb{Z}/m\mathbb{Z} \times\mathbb{Z}/n\mathbb{Z}$ is cyclic.
105139
Verification of Proof that a nonabelian group G of order pq where p and q are primes has a trivial center
14890
Let $G=\langle a\rangle$ be a cyclic group of order $n$. Show that, for every divisor $d$ of $n$, there exists a subgroup of $G$ whose order is $d$.
43580
Let $p$ be a prime number and $G$ a non abelian group or order $p^3$. Prove that $Z(G) = [G,G]$.
108425
Problem on abelian group
71420
Prove that a group is abelian.