Let $G$ be a group and $a,b∈G$. Suppose order of $(ab)$ in $G$ is $n$. Show that $(ba)^n=e$ Prove that the order of $(ba)$ in $G$ is $n$.
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 $G$
33593
An example of a non-abelian group $G$ where, for all $a,b\in G$, the equality $(ab)^{n}=a^{n}b^{n}$ holds for two consecutive integers $n$
140334
Let $G$ be a group and $H$ be a subgroup of $G$ of index $2$ .Then show that if $a,b\notin H$ then $ab\in H$.
150958
Group theory question. If $m$ and $n$ are relatively prime, and $a$, $b$ belong to group and $a^m = b^m$ and $a^n = b^n$, how would one prove $a = b$?
62035
Order of automorphism group of cyclic group
121142
Take $G$ to be a group of order $600$. Prove that for any element a in $G$ there exist a $b$ in $G$ such that $a = b^7$
117468
$o(g^{k})=\frac{o(g)}{\gcd(o(g),k)}$ order of a power of a group element.
37061
The order of $ab$ when $a,b$ commute
83654
Order of an element in a finite cyclic group
37913
Prove that $ AB$ is a subgroup of order $mn$ if $m$ and $n$ are relatively prime.
158074
Cyclic group $G$ with generator $ab$.
90437
Let $G = \langle a\rangle$ a cyclic group of order m. Prove that G is isomorphic with $Z_m$.
79055
Orders of a symmetric group
143550
Order of product of two elements in an abelian group
79933
Let $G$ be a group and $c\in G$. Define $*$ on $G$ by: $x*y=xdy$ for all $x,y\in G$, where $d$ is the inverse of $c$. Prove $(G,*)$ is a group.
150156
Proving that $S_n$ has order $n!$