Let $G$ be a group of order $n$. $G$ has one and only one subgroup of order of every divisor of $n$. Then G is cyclic.
40513
A finite group which has a unique subgroup of order $d$ for each $d\mid n$.
23394
If $G/Z(G)$ is cyclic, then $G$ is abelian
21673
If $|G| = p^n$ then $G$ has a subgroup of order $p^m$ for all $0\le m <n.$
147517
Any group of order $n$ satisfying $\gcd (n, \varphi(n)) =1$ is cyclic
112157
If $g$ is the generator of a group $G$, order $n$, when is $g^k$ a generator?
76446
Why are cyclic groups of order $p$ (prime) unique up to isomorphism?
61379
Let $G$ be abelian, $H$ and $K$ subgroups of orders $n$, $m$. Then G has subgroup of order $\operatorname{lcm}(n,m)$.
106653
How many generator has a cyclic group of order n?
74586
Prove that every group of order $4$ is abelian
108830
Let $G$ be a group of order $2p$ , where $p$ is a prime greater than $2$. Then, G is isomorphic to $\mathbb{Z}_{2p}$ or $D_p$
90437
Let $G = \langle a\rangle$ a cyclic group of order m. Prove that G is isomorphic with $Z_m$.
114187
In a finite cyclic group, the generator has the order of the group
22301
If $G$ has no proper subgroup, then $G$ is cyclic of prime order
84823
When is the group of units in $\mathbb{Z}_n$ cyclic?
36887
Suppose $G$ is a group of order 4. Show either $G$ is cyclic or $x^2=e$.
50762
If $S_n$ has cyclic subgroups of these orders or not?
142894
Showing that a group of order $pq$ is cyclic if it has normal subgroups of order $p$ and $q$