I think this proof is a bit sloopy, but the idea works right? If $G/Z(G)$ is cyclic, then $G$ is abelian.
23394
If $G/Z(G)$ is cyclic, then $G$ is abelian
65900
Show that a finite group with certain automorphism is abelian
59128
If $|\lbrace g \in G: \pi (g)=g^{-1} \rbrace|>\frac{3|G|}{4}$, then $G$ is an abelian group.
43783
Abelian group (Commutative group)
144065
Proving $\phi(G)$ is cyclic if $\phi$ : $G \to H$ is group homomorphism and $G$ is cyclic
98145
If $G$ is cyclic then $G/H$ is cyclic?
127978
Another point of view that $\mathbb{Z}/m\mathbb{Z} \times\mathbb{Z}/n\mathbb{Z}$ is cyclic.
125559
In a group $G$, if for all $a,b,c\in G$, $ab=ca\Rightarrow b=c$, then $G$ is abelian
5223
$\mathbb{Z}_m \times \mathbb Z_n$ is cyclic if and only if $\gcd(m,n)=1$
87882
If $N \lhd G$ is abelian and $G/N$ is abelian then $G$ is abelian
108213
$N_1,N_2,N_3 \unlhd G, N_i\cap N_j =\{e\}, G = N_iN_j$. Want to show that $G$ is abelian, $N_i$ are isomorphic.
117206
$|G|$ and $|{\rm Aut}(G)|$ are relatively prime $\Rightarrow$ $G$ should be cyclic?
98647
$G$ is Topological $\implies$ $\pi_1(G,e)$ is Abelian
84
For what $n$ is $U_n$ cyclic?
1364
Let $G$ be a group in which $a^2=e$ for all elements of $a$ of $G$. Show 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 $G$
69046
Proof of $\mathbb{Q}$ is not cyclic