Products of disjoint cycles. Explanation please.

[tamintl]

Hey all,

Currently trying to revise products of disjoint cycles. I am struggling to get my head around the idea however. Its strange, i can generate disjoint cycles easily however i can't quite see the product of cycles. Mabye I've missed something.

eg of what i don't quite get.

What is the product of
(1,2,3)(1,2)

My textbook says the answer is (1,3) but i can't understand how this is?

Please shed some light!

Kind regards
Tamintl

 

[I like Serena]

The cycles are applied right to left.

The cycle (1 2) means that 1→2→1.
The cycle (1 2 3) means 1→2→3→1.

So 1→2 by the second cycle and then 2→3 by the first cycle.
In the same way 3→3 by the second cycle, and then 3→1 by the first cycle.
Finally 2→1→2, so 2 does not change.

So the combination of the 2 cycles is 1→3→1 or (1 3).

Cheers!

 

[tiny-tim]

Welcome to PF!

Hey Tamintl! Welcome to PF!

So far as I know, you have to do it for each element separately (starting at the right).

So 2 goes to 1 and then 1 goes to 2.

3 goes to 3 and then 3 goes to 1.

1 goes to 2 and then 2 goes to 3.

That makes (1,3). ​

 

[Stephen Tashi]

Hey all,

Currently trying to revise products of disjoint cycles. I am struggling to get my head around the idea however. Its strange, i can generate disjoint cycles easily however i can't quite see the product of cycles. Mabye I've missed something.

eg of what i don't quite get.

What is the product of
(1,2,3)(1,2)

My textbook says the answer is (1,3) but i can't understand how this is?

Please shed some light!

Kind regards
Tamintl

(1,2) abbreviates the function g(x) defined as follows:
g(1) = 2
g(2)= 1
g(x) = x for all other values of x

(1,2,3) abbreviates the function f(x) defined as follows:
f(1) = 2
f(2) = 3
f(3) = 1
f(x) = x for all other values of x

The product (1,2,3)(1,2) abbreviates the composition of functions h(x) = f(g(x))

h(1) = f(g(1)) = f(2) = 3
h(2) = f(g(2)) = f(1) = 2
h(3) = f(g(3)) = f(3) = 1

This can be summarized as
h(1) = 3
h(3) = 1
h(x) = x for other values of x

Hence h(x) can be abbreviated as (1,3)

Of course, you can find simpler ways to visualize the multiplication so you don't have to write out the process in that much detail. Start with an ordered set (a,b,c) and apply the permutations to it and note the relative postions of the letters when you finish. Express how the letters are displaced as a product of cycles.

 

[tamintl]

Wow thanks guys.. i see it now! I didnt realize you visualised the disjoints as permutations on their own, and then link them together


Thanks

 

[I like Serena]

I'm impressed that you got a cycle of 3 people answering the same question at the same time.
Especially tiny-tim who achieved to publish his post in the same minute as I did!