Permutations with no common symbols

Question

Let $\sigma$ and $\tau$ be permutations with no common symbols, and $\sigma \tau = e$, then $\sigma = \tau = e$. Should I denote the permutations as cycles or transpositions? How to write them with the fact that they does not have common symbols? What I really need is the way to write this so I get the proof.

Answer 1

Presumably "no common symbols" refers to the representation of $\sigma$ and $\tau$ as disjoint cycles, omitting all $1$-cycles (i.e., elements that get mapped to themselves).

Now, given $\sigma\tau = e$, take some element $a$ in the set, and consider how $\tau$ acts on it. There are two cases: either $\tau$ fixes $a$, or $a$ appears in a cycle of length at least $2$. Can you continue from here?

Note that it would be possible to describe $\sigma$ and $\tau$ with variables, but that would quickly get cumbersome, e.g., $\sigma = (a_{11}a_{12}\cdots a_{1k_1})\cdots$, etc. It's easier here to just use words, as in the paragraph above.