[[Group order]]
# Relationship between $\abs{ab}$ and $\abs a \abs b$

Given a finite group $G$, and elements $a,b \in G$ such that $ab = ba$,
then $\abs{ab}$ divides $\abs a \abs b$. #m/thm/group 

> [!check]- Proof
> Let $\abs{a} = m$ and $\abs b = n$.
> Then $(ab)^{mn} = (a^m)^n(b^n)^m = e$ which implies $\abs{ab}$ divides $mn$ by [[Criterion for 𝑎ⁱ = 𝑎ʲ in a group#Corollary]].
> <span class="QED"/>


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#state/tidy | #lang/en | #SemBr