So I have been trying to understand permutations, transpositions, parity...
And something isn't clicking.
In particular, say we have a permutation:
$$\sigma = \begin{pmatrix} 1 & 2 & 3 & 4 & 5\\ 2 & 5 & 1 & 3 & 4 \end{pmatrix} $$
One possible cycle notation for this is: (1,2,5,4,3)
This can be created naively by just following the entries in the table.
However (it may be for this example but it's not in general) a cycle notation for a permutation need not be unqiue. What rules can you use to generate a cycle notation for a given permutation aside from naively following the entires on teh table? i.e What algebraic rules can be used to generate a cycle from the original permutation table and/or from an already existing cycle?
I assume the above would also explain how to execute transpositions, in which case I need to understand only one thing
Given the transpositions: $(uw)(uv)$ you can generate the transpositions $(uw)(wv)$ and this is supposed to keep the permutation intact. Why is that?