@DanielFischer With permutations that aren't transpositions you mean that we could have more than two numbers, for example ( 1 2 3) ?
If it suffices to look at $\tau$ of the form $(1\,m)$ because $S_n$ is generated by transpositions of that form, why isn't g also of that form, given that it is also an element of $S_n$ ?
If we have $g = (a_0\, a_1\, \dotsc\, a_{k-1})$ can it be that $a_i=1$ for some i?