If $G$ is abelian, $|G|=p^k$ for $k\in\mathbb{Z}$, and $G$ has a unique subgroup of order $p$, then $G$ is cyclic.
54453
Finite abelian $p$-group with only one subgroup size $p$ is cyclic
90811
If $[G' : G'']\leq p^2$, then $G'$ is abelian.
100605
let $G$ to be finite abelian group of order O(G), let n to be prime number and (O(G),n)=1 prove that $g=x^n$ for any $ x \in G$
45727
If $G$ contains a normal subgroup $H \cong \mathbb{Z_2}$ such that $G/H$ is infinite cyclic, then $G \cong \mathbb{Z} \times \mathbb{Z_2}$
138406
If N and every subgroup of N is normal in G then G/N is abelian .
7967
Prove that the order of the cyclic subgroup $\langle g^k\rangle $ is $n/{\operatorname{gcd}(n,k)}$
102146
If a group mod its commutator subgroup is cyclic, then the group is abelian?
106577
If G is not abelian with order $p^3$ then $G'=Z(G)$
26624
Prove that is group is abelian
147517
Any group of order $n$ satisfying $\gcd (n, \varphi(n)) =1$ is cyclic
73211
Cyclic subgroups dealing with prime numbers Question
57887
When are the product of cyclic groups also cyclic?
34902
If $H$ is a cyclic subgroup of $G$ and $H$ is normal in $G$, then every subgoup of $H$ is normal in $G$.
68201
Cyclic group of order $n$
145112
Show that the multiplicative group $(\mathbb{Z}/p\mathbb{Z})^*$ is cyclic
94239
Proof that if group $G/Z(G)$ is cyclic, then $G$ is commutative
151619
A finite abelian group containing a non-trivial subgroup which lies in every non-trivial subgroup is cyclic