A group $G$ of order $p^n$ is cyclic iff it is abelian having a unique subgroup of order $p$.
134820
$p$ prime, Group of order $p^n$ is cyclic iff it is an abelian group having a unique subgroup of order $p$
106123
Is there a finite abelian group $G$ such that $\textrm{Aut}(G)$ is abelian but $G$ is not cyclic?
50762
If $S_n$ has cyclic subgroups of these orders or not?
151025
If G is a group of order n=35, then it is cyclic
57464
If $G$ is a group show that if $(a \cdot b)^2 = a^2 \cdot b^2$ then $G$ must be abelian.
38853
Cyclic group of prime order
21673
If $|G| = p^n$ then $G$ has a subgroup of order $p^m$ for all $0\le m <n.$
36887
Suppose $G$ is a group of order 4. Show either $G$ is cyclic or $x^2=e$.
76504
Non-trivial cyclic group with unique generator
73402
Showing that $(\mathbb{Z}/p^{a}\mathbb{Z})^*$ is a cyclic group
101602
consider the group $G=\mathbb Q/\mathbb Z$. For $n>0$, is there a cyclic subgroup of order n
129061
How to say that any group of order $pqr$ is cyclic with a provided relation between $p$, $q$, $r$, where they are all distinct primes.
106577
If G is not abelian with order $p^3$ then $G'=Z(G)$
40513
A finite group which has a unique subgroup of order $d$ for each $d\mid n$.
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.
126893
A group of order 2p (p prime) and other conditions - prove abelian.
7967
Prove that the order of the cyclic subgroup $\langle g^k\rangle $ is $n/{\operatorname{gcd}(n,k)}$