Prime Notation in Transpositions

Question

My textbook has the following problem:

Consider the group $S_n.$ Prove that if $\sigma$ and $\tau$ are any pair of distinct transpositions such that $\tau(1)\neq1$ then there exists $\sigma'$,$\tau'$ such that $\tau'(1)=1$, $\sigma'(1)\neq1$, and $$\sigma\tau=\sigma'\tau'.$$

What would the prime symbol mean in this context?

Answer 1

$\sigma'$ and $\tau'$ are simply two other permutations, which you have to find satisfying the given properties. That a prime is used to differentiate it from the un-primed version may suggest that the two share some structure.

Answer 2

You might equivalently write:

$\forall \sigma,\tau\in S_n$ such that $\sigma,\tau$ are distinct transpositions and $\tau(1)\ne 1$, $\exists \rho,\theta\in S_n$ such that $\theta(1)=1$, $\rho(1)\ne 1$ and $\sigma\tau=\rho\theta$. Prime symbol just helps in recalling that $\rho,\theta$ depend on $\sigma,\tau$.