Finite abelian group A cyclic iff for every $n\in \mathbb{N}$, $|\{a\in A \mid a^n = 1\}| \leq n$
35664
If $x^m=e$ has at most $m$ solutions for any $m\in \mathbb{N}$, then $G$ is cyclic
90239
$\mathbb{Z} \times \mathbb{Z}$ is cyclic.
129485
A subgroup of a cyclic group is cyclic - Understanding Proof
116125
A group $G$ with $G'$ abelian and every abelian normal subgroup finite
7079
ord $(b)|\max\{\text{ord}(g)|g\in G\}$ for all $b\in G\,$ a finite abelian group
48393
Abelian groups of order n.
81543
Finite abelian group contains an element with order equal to the lcm of the orders of its elements
150585
Group with $a=a^{-1}$ for all $a\in G$ is abelian
68733
If $G/Z(G)$ is cyclic then $G$ is abelian – what is the point?
135039
${\rm Aut}(G)$ is cyclic $\implies G$ is abelian
132725
Is $\mathbb{Z}/\langle n\rangle$ cyclic and of order $n$?
65900
Show that a finite group with certain automorphism is abelian
5297
If G is a finite abelian group and $a_1,...,a_n$ are all its elements, show that $x=a_1a_2a_3...a_n$must satisfy $x^2=e$.
32620
Is every group of order $21$ cyclic?
139835
Every Finite Group is Isomorphic to a Subgroup of $A_n$
49768
If Gal(K,Q) is abelian then |Gal(K,Q)|=n
147517
Any group of order $n$ satisfying $\gcd (n, \varphi(n)) =1$ is cyclic