If $G$ is a group, $x,y, \in G$ and $xy =yx$ then prove that $|xy| = \operatorname{lcm}\{|x|,|y|\}$ given that $|x|$ and $|y|$ are coprime
11509
Relationship between $\operatorname{ord}(ab), \operatorname{ord}(a)$, and $\operatorname{ord}(b)$
154959
Proof of associativity of GCD?
59462
Prove that $n!=\prod_{k=1}^n \operatorname{lcm}(1,2,...,\lfloor n/k \rfloor)$ for any $n \in \mathbb N$
121156
Prove that: $\gcd[a,b,c]=\frac{abc.\operatorname{lcm}(a,b,c)}{\operatorname{lcm}(a,b)\operatorname{lcm}(a,c)\operatorname{lcm}(b,c)}$
132587
Prove that the product of coprimes must be congruent to $-1$ or $1 \pmod{m}$
125737
GCD( LCM(a,b) , LCM(a,c) ) = LCM(b, GCD(b,c)) , why?
7967
Prove that the order of the cyclic subgroup $\langle g^k\rangle $ is $n/{\operatorname{gcd}(n,k)}$
81543
Finite abelian group contains an element with order equal to the lcm of the orders of its elements
20679
Prove that if $g^2=e$ for all $g$ in $G$ then $G$ is Abelian.
94239
Proof that if group $G/Z(G)$ is cyclic, then $G$ is commutative
141683
$ |G_1 |$ and $|G_2 | $ are coprime. Show that $K = H_1 \times H_2$
148108
Let $L=\operatorname{lcm}(o(a), o(b))$ and prove that $(ab)^L =e$.
106927
How do I prove $[G:H\cap K]\leq [G:H][G:K]$?
14890
Let $G=\langle a\rangle$ be a cyclic group of order $n$. Show that, for every divisor $d$ of $n$, there exists a subgroup of $G$ whose order is $d$.
48452
Proof that $\gcd$ divides $\operatorname{lcm}$
2208
$\gcd(a,b,c)=\gcd(\gcd(a,b),c)\,$ [GCD Associative Law]
83776
Is there a direct proof of this lcm identity?