If $G$ is cyclic with order $N$ then $g^n$ has order $\frac{N}{\gcd(n,N)}$
7967
Prove that the order of the cyclic subgroup $\langle g^k\rangle $ is $n/{\operatorname{gcd}(n,k)}$
83654
Order of an element in a finite cyclic group
132475
Simple method the show that a group of order 15 is cyclic
141890
$G$ is a Group of order $pq$, if $G$ has exactly one subgroup of order $p$ and another with order $q$, then $G$ is cyclic ($p,q$ are prime)
75127
Group $G$ of order $p^2$: $\;G\cong \mathbb Z_{p^2}$ or $G\cong \mathbb Z_p \times \mathbb Z_p$
2208
$\gcd(a,b,c)=\gcd(\gcd(a,b),c)\,$ [GCD Associative Law]
124309
$g\in G$ maximal order in $G$ abelian then $G=\left\oplus H$
45156
Every group of order $35$ is cyclic?
110313
Prove that $\gcd(f(x),g(x)) = 1$.
35664
If $x^m=e$ has at most $m$ solutions for any $m\in \mathbb{N}$, then $G$ is cyclic
3703
Order of finite fields is $p^n$
140471
Is $ G \cong G/N \times N$?
15387
Order of $GL(n, \mathbb Z_p)$
5069
$GL_n(\mathbb F_q)$ has an element of order $q^n-1$
44152
If $G$ has only one subgroup of order $n$, then that Subgroup is Normal
6012
Prove that $\gcd(a^n - 1, a^m - 1) = a^{\gcd(n, m)} - 1$
152662
Proving that any finite cyclic group of order $n$ is isomorphic to $$