||durka42: I tried that, but a lot of it's built on orbits and transpositions of the form (k (k+1))|
Our building blocks are different. We use orbits of the form (1 k)
I've shown three identities which are really all you need:
Let SE be the set of SE.
We consider strings of SE and call this set "SE*".
Let Ø be the empty string.
∀ a ∈ SE : aa = Ø
We can observe that each SE is a function and a permutation, and that a string can be interpreted as a composition in the normal order.
I will now show that for r_k ∈ SE, r_1 r_2 ... r_k r_1 = r_2 ... r_k r_1 r_2
i ≠ j <=> r_i ≠ r_j
We call this operation a rotation.
As a concrete example se te ve se broda = te ve se te broda.
You can verify yourself, if you wish
The basic idea is that we're using the 1st position as a sort of swap space to build an orbit.
Whenever we do the first swap we store the first position that spot then we store each successive position in the spot that follows.
Finally, we restore the 1 position and swap the final value from the chain into the starting position, completing the orbit.
Let's say we want the place structure 1 4 2 3 5.
That is 4 -> 2 2 -> 3 and 3 -> 4
In a permutation group, we call such a permutation an "orbit"
We would write it this way: (4 2 3) or equivalently (2 3 4)
We reverse it, and translate it into conversions: «ve te se», then tack on a «se» or a «ve» to restore the position to get se ve te se or ve te se ve.