Basic group theory 01

[Grigorios Kostakos]

Let a group $G$ and $x,a,b\in{G}$, such that: \[x=a\,b=b\,a \quad {\text{and}} \quad a^{p}=1=b^{q}\,,\] where $p$ and $q$ are relative prime numbers. Prove that exists relative prime numbers $n$ and $m$, such that \[a=x^{n}\quad {\text{and}} \quad b=x^{m}\,.\]

[dr.tasos]

So we got $ \exists k , l \in \mathbb{Z} $ such that $ kp+lq=1 $. So $ kp$ and $lq $ are relatively prime (Bezout ) .


$$ (ab)^{lq}=a^{1-kp}b^{lq}=a a^{-kp}=a=x^{lq}\\
(ab)^{kp}=a^{kp}b^{1-ql}=b=x^{pk} $$