Groups of permutations and cyclic groups

[yaganon]

1: Is a group of permutations basically the same as a group of functions? As far as I know, they have the same properties: associativity, identity function, and inverses.

2: I don't understand how you convert cyclic groups into product of disjoint cycles.
A cyclic group (a b c d ... z) := a->b, b->c, c->d, d->e ... y->z, z->a
In the book, it shows that (0 3 6) o (2 7) o (4 8) o (0 4 7 2 6) o (1 8) = (0 8 1 4 2) o (3 6)

How do you get there?

thanks

 

[Martin Rattigan]

1. Yes, it's a group of 1-1 functions from the set of elements being permuted (or possibly a superset) onto itself.

2. You don't, you may express individual members of a cyclic group of permutations as the product of disjoint cycles. In your example each cycle goes from left to right but the products are evaluated from right to left. The direction of evaluation could be different in a different book, but the cycles will usually go from left to right I believe.

So for example:

1--(18)--> 8 --(0 4 7 2 6)--> 8--(4 8)--> 4 --(2 7)--> 4 --(0 3 6)--> 4

Which is why 4 follows 1 in the cycle (0 8 1 4 2) on the rhs. Other elements similarly.

If g=(abc...z) then g² would be (abc...z)(abc...z) evaluated the same way, i.e. (ace...y)(bdf...z) and so on.

 

[yaganon]

OK, I kind of get it. So you always start with 0, and the last element ends with 2 because two maps to 0 through the series of functions, completing the cycle. What if there isn't a zero? Also, when you're done with a cycle, how do you start the next one, do you choose the next smallest element that's left? which is why it's (3 6) and not (6 3)?

 

[Martin Rattigan]

It doesn't matter what you start with. All that happens is your cycles can appear shifted round if you start with something different.

After finding 1->4 I woulve tracked 4 etc. Then my cycle would appear as (14208) instead of (08142) on thr rhs. But it's the same cycle either way.

 

[Martin Rattigan]

Similarly you can just pick anything you haven't already done for the next cycle. It helps to take them in some sort of order so you don't forget a cycle altogether, but it's up to you.

 

[Martin Rattigan]

That is to say (63) will do just as well as (36). It's exactly the same mapping.