Intersection of cyclic subgroups: $(x^m) \cap (x^n) = (x^{lcm(m,n)})$
2626
Show $\langle a^m \rangle \cap \langle a^n \rangle = \langle a^{\operatorname{lcm}(m, n)}\rangle$
152071
Group $G$ is cyclic $\iff$ every subgroup of $G$ has the form $G^k$, where $G^k=\{g^k\mid g\in G\}$
11600
$\gcd(a,\operatorname{lcm}(b,c))=\operatorname{lcm}(\gcd(a,b),\gcd(a,c))$
101602
consider the group $G=\mathbb Q/\mathbb Z$. For $n>0$, is there a cyclic subgroup of order n
58622
Intersection points of $2^x$ and $x^2$
73174
Prove $KN\cong K\times N$ if $K\cap N=\{e\}$
102146
If a group mod its commutator subgroup is cyclic, then the group is abelian?
27215
Intersection of the $p$-sylow and $q$-sylow subgroups of group $G$
68201
Cyclic group of order $n$
9393
Order of elements is lcm-closed in abelian groups
139833
Prove that $\gcd(M, N)\times \mbox{lcm}(M, N) = M \times N$.
107716
Show that $G$ is cyclic
25744
Multiplication in Permutation Groups Written in Cyclic Notation
43728
Proof via Group Theory : $\mathrm{lcm}(a,b) \cdot \gcd(a,b) = |ab|$
145112
Show that the multiplicative group $(\mathbb{Z}/p\mathbb{Z})^*$ is cyclic
33945
Showing that $|N \cap Z(G)| > 1$ for normal subgroups of p-groups
150042
Prove $x^{-1}Hx$ is a subgroup of $G$