How to prove that G is a cyclic Group?
107716
Show that $G$ 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.
82037
Suppose $G$ is a group generated by elements $x$ and $y$ where $xy^2 = y^3x$ and $yx^3 = x^2y$ What can you prove about $G$?
59533
If $\gcd(m,n)=1$, then $\mathbb{Z}_n \times \mathbb{Z}_m$ is cyclic.
129485
A subgroup of a cyclic group is cyclic - Understanding Proof
53357
Cyclic Group Generators of Order $n$
17077
Prove or disprove: If $H$ and $G/H$ are cyclic, then $G$ is cyclic.
38853
Cyclic group of prime order
147517
Any group of order $n$ satisfying $\gcd (n, \varphi(n)) =1$ is cyclic
36040
A Group Having a Cyclic Sylow 2-Subgroup Has a Normal Subgroup.
132127
Proving a direct product of groups is a group
84823
When is the group of units in $\mathbb{Z}_n$ 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.
7967
Prove that the order of the cyclic subgroup $\langle g^k\rangle $ is $n/{\operatorname{gcd}(n,k)}$
47179
Prove that $[N, G']=1$ when $N$ is a nilpotent subgroup of a group $G$?
73402
Showing that $(\mathbb{Z}/p^{a}\mathbb{Z})^*$ is a cyclic group
144065
Proving $\phi(G)$ is cyclic if $\phi$ : $G \to H$ is group homomorphism and $G$ is cyclic