If $(G, \times)$ is a cylic group and of order $n$,then for $m<n$ the order of $g^m$ is $\frac{n}{\gcd\left(n,m\right)}$.
5789
$ d = \gcd(a,b)\Rightarrow\ \gcd(a/d,b/d) = 1$
33300
Showing order of $Z (G) =1$ or $pq$
139833
Prove that $\gcd(M, N)\times \mbox{lcm}(M, N) = M \times N$.
50706
$|G|=p(p+1)$ for $p$ prime, then $G$ has a normal subgroup of order $p$ or $p+1$
135808
Show that there is a positive integer $n$ such that $G \cong \ker(\phi^{n}) \times \phi^{n}(G)$
123796
Order of automorphism group of a $p$-group is divisible by $p$.
8841
A $p$-group of order $p^n$ has a normal subgroup of order $p^k$ for each $0\le k \le n$
134820
$p$ prime, Group of order $p^n$ is cyclic iff it is an abelian group having a unique subgroup of order $p$
61379
Let $G$ be abelian, $H$ and $K$ subgroups of orders $n$, $m$. Then G has subgroup of order $\operatorname{lcm}(n,m)$.
152481
If order of group is $p^2$, where $p$ is prime, how can you deduce $G$ is isomorphic to $C_{p^2}$ or $C_p \times C_p$?
78852
If $\gcd(m,n) =1, $ then $m\mathbb{Z}/ mn\mathbb{Z} \cong \mathbb{Z}/(n)$
148904
Groups of order $p^3$
81116
$gh = hg, \ \gcd(|g|, |h|) = 1\Rightarrow|gh| = |g||h|\ \ (|a|$ = order of element $a)$
30003
Proving that Order for N is Anti-symmetric
43058
If $n > 2$, prove that the order of the multiplicative group of units modulo n, $U_n$, is even.
133915
In which case $M_1 \times N \cong M_2 \times N \Rightarrow M_1 \cong M_2$ is true?
157841
Existence of a non-abelian group of order $p^n$.