$g_{1},g_{2} \in G$, then if $g_{1}g_{2}$ has finite order then we have to prove that $g_{2}g_{1}$ also has finite order too.
98331
Show that if $ab$ has finite order $n$, then $ba$ also has order $n$. - Fraleigh p. 47 6.46.
134376
Let $G$ be a nonabelian group. Let $a \in G$ have order $4$ and let $b \in G$ have order $3$. Then the order of the element $ab$ in $G$
1577
Order of nontrivial elements is 2 implies Abelian group
158282
Finite order of an element $f(g)$ of a homomorphism $f$
62077
Prove that if $W_1\subseteq V$ finite-dimensional, then there is $W_2\subseteq V$ such that $V=W_1\oplus W_2$
79055
Orders of a symmetric group
121423
Prove that every element of a finite group has an order
6115
Show that $GL_n(\mathbb{F})$ is a finite group iff $\mathbb{F}$ has a finite number of elements
23885
Structure of a group, $G$, of order $pq$ where $p, q$ are prime.
85129
If $H_1, H_2\leq G$ are such that $H_1\cong H_2$ then $G/H_1\cong G/H_2$?
99205
Group of order $p^k q$ where $p$ and $q$ are distinct primes - how to prove this property?
122970
A question on $p$-groups, and order of its commutator subgroup.
113357
Groups of order $pqr$ and their normal subgroups
141683
$ |G_1 |$ and $|G_2 | $ are coprime. Show that $K = H_1 \times H_2$
101735
A finite $p$-group cannot be simple unless it has order $p$
104080
Group with order $pq$ has subgroups of order $p$ and $q$
100605
let $G$ to be finite abelian group of order O(G), let n to be prime number and (O(G),n)=1 prove that $g=x^n$ for any $ x \in G$