Show that $\tau_1\tau_2$ of order $2$ or $3$, given that $\tau_1$ and $\tau_2$ are distinct transpositions.
I know that every cycle can be factored as a product of transpositions. But I'm not sure how to prove the order of $2$ or $3$.
Show that $\tau_1\tau_2$ of order $2$ or $3$, given that $\tau_1$ and $\tau_2$ are distinct transpositions.
I know that every cycle can be factored as a product of transpositions. But I'm not sure how to prove the order of $2$ or $3$.
Write $\tau_1 = (a,b)$, $\tau_2=(r,s)$.
We have two cases:
$\{a,b\}\cap\{r,s\} = \emptyset$. What happens to the product then?
$\{a,b\}\cap\{r,s\}\neq\emptyset$. Then the two share exactly one element (Why?) so we can write $\tau_1 = (a,b)$, $\tau_2=(a,s)$. What is the product then?