Marco Fulvio's Dust

Permutations

matan-matika

Figuring out how to arrange things is pretty important.

Like, if we have the letters {A,B,C}, the six ways to arrange them are:
ABC
ACB
BAC
BCA
CAB
CBA

And we can say more interesting things about them (e.g. Combinatorics) another great extension is when we get dynamic

Like, if we go from ABC to ACB, and back…

We can abstract away from needing to use individual letters, and say these are both “switching the 2^nd and 3^rd elements,” and it is the same thing both times.

Each of these switches can be more complicated than that, like going from ABCDE to EDACB is really just 1->3->4->2->5->1, and we can do it 5 times and cycle back to the start

We can also have two switches happening at once, like 1->2->3->1 and 4->5->4, and this cycles through 6 times to get to the start.

Then, let’s extend this a bit further.

First, let’s first get a better notation, and use (1 2 3) for what I called 1->2->3->1 before.

Let’s show how we can turn these permutations into a group.

Then, let’s say the identity is just keeping things the same, and call it id.

And, this repeating thing can be extended into making the group combiner: doing one permutation and then the other. For various historical reasons, the combination of permutation A and then permutation B is B·A.

This is closed, because permuting all the things and then permuting them again still keeps 1 of all the elements in an order.

Inverses exist, because you just need to put everything from the new position into the old position to reverse it.

Associativity will be left as an exercise to the reader (read: I don’t want to prove it)

See, that’s what the app is perfect for.

An account of the Crustacea of Norway, with short descriptions and figures of all the species.

Permutations