Is there a "larger picture" from which we can see that $G$ is abelian if $G/Z(G)$ is cyclic?
68733
If $G/Z(G)$ is cyclic then $G$ is abelian – what is the point?
57421
If $\forall x \in G$ where G is a group $x=x^{-1}$ then how is this group abelian?
102146
If a group mod its commutator subgroup is cyclic, then the group is abelian?
87882
If $N \lhd G$ is abelian and $G/N$ is abelian then $G$ is abelian
59533
If $\gcd(m,n)=1$, then $\mathbb{Z}_n \times \mathbb{Z}_m$ is cyclic.
74964
Showing Galois Group is Abelian
106809
$f: G→G$ defined by $f(x) =x^2$ is a homomorphism if and only if $G$ is abelian.
20679
Prove that if $g^2=e$ for all $g$ in $G$ then $G$ is Abelian.
40993
Let $f:G\to G'$ be a surjective homomorphism. Prove that if $G$ is cyclic, then $G'$ is cyclic.
106123
Is there a finite abelian group $G$ such that $\textrm{Aut}(G)$ is abelian but $G$ is not cyclic?
119512
If H ≤ Z(G) ≤ G, where G is a finite group,Z(G) is its center, and (G:H) = p for some prime p, then G is abelian.
31511
A finite abelian group that does not contain a subgroup isomorphic to $\mathbb Z_p\oplus\mathbb Z_p$, for any prime $p$, is cyclic.
135188
Show that if $\vert G\vert = pq$, then either $G$ is abelian or $Z(G) = \{ e\}$.
57464
If $G$ is a group show that if $(a \cdot b)^2 = a^2 \cdot b^2$ then $G$ must be abelian.
33025
What is the order of $(\mathbb{Z} \oplus \mathbb{Z})/ \langle (2,2) \rangle$ and is it cyclic?
156429
If $G$ is a group whereby $(a\cdot b)^{i} =a^i\cdot b^i$ for three consecutive integers $i$ for all $a, b \in G$, show $G$ is abelian.
45760
if G is generated by {a,b} and ab=ba, then prove G is Abelian