00:01
It would be then two permutations, sigma and tau, and s11.
00:07
We want to write sigma as a product of disjoint cycles.
00:11
So i just start at one and see what's on me.
00:14
So one goes to five.
00:16
Five goes to four.
00:19
Four goes back to one.
00:21
So we're done.
00:21
We did one for and four.
00:23
Okay.
00:24
Next one, two goes to seven.
00:27
Seven goes to six.
00:29
Six goes to nine.
00:31
Nine goes to eleven.
00:33
11 goes to 3, 3 goes back to 2, so we're done.
00:39
We did 2, 7, 6, 9, 11, 3.
00:46
And then lastly 8 goes to 10 and 10 goes back to 8.
00:51
Okay, so those are our disjoint cycles.
00:53
Okay, determine the order of tau, so that's the power of tau.
00:57
The smallest power of tau for which we're going to get the identity permutation.
01:02
Okay, so that's going to be the least common multiple, of the lengths of each disjoint cycle.
01:10
So this is the least common multiple of four, five, two, and three.
01:20
Oh, sorry, these are not disjoint yet.
01:26
Okay, so before i determine the order of tell, i need to write this probably disjoint cycles.
01:31
So this is going to be one goes to four, four then goes nowhere, nowhere, four goes to six, six, six, six, six, six goes back to one, so that's it.
01:45
Okay, next two, two goes nowhere, nowhere, two goes to three, three goes nowhere, nowhere, nowhere, nowhere, three goes to eight, nowhere, nowhere, nowhere, eight goes to ten, nowhere, nowhere, nowhere, nowhere, nowhere, ten goes to seven, nowhere, nowhere, nowhere, nowhere, seven goes to two, nowhere, so that's it.
02:11
We'll get one two three i need four next four goes to nine nine goes to eleven then nowhere okay 11 goes to nine and then nowhere nine goes to one then one goes to five well nowhere nowhere nowhere nowhere five goes to four so that's it here okay one two three four five six seven eight nine ten eleven okay so got them all so now i can do the lcm of the disjoint cycles.
02:46
So length is 2, 5, and 4.
02:52
Okay, so at least count multiple is 20.
02:55
Okay, so the order of tau is 20.
02:58
Okay, next we want to write tau inverse as a product of disjoint cycles.
03:03
So if 1 goes to 6, then 6 needs to go to 1.
03:12
Okay, here we'll just flip it around.
03:15
So if 10 goes to 7, 7 needs to go to 10, 8 goes to 10, so 10 goes to 8, 3 goes to 8, so 8 goes to 3, and then 2 goes to 3, so 3's got to go to 2, and then 7 goes to 2, so 2 goes back to 7, and then reorder 5, 9, 11, 4.
03:39
Okay, so if you multiply these together, we should get the identity, right? okay, represent sigma, tau as a product of disjoint cycles, and an array notation.
03:48
Okay, so sigma, tau.
03:52
So i'm applying tau first and then sigma.
03:55
Okay, so sigma and then tau.
03:58
So we have 154.
04:00
I'm doing sigma here.
04:02
154, 27, 6, 9, 11, 3, 8, 10.
04:09
Multiply with tau.
04:11
I'll give myself in the phone here.
04:16
Then tau was 1, 6, 2, 3, 8, 10, 7.
04:22
Four, eleven, nine, five.
04:27
Okay, so one goes nowhere, nowhere, one goes to six, nowhere, six goes to nine, nowhere.
04:36
Nine goes to five, nowhere, nowhere, nowhere, nowhere, five goes to four.
04:42
Okay, four goes to eleven, nowhere, nowhere, nowhere, eleven goes to three, nowhere.
04:48
Three, three goes to eight, nowhere, it goes to ten, nowhere, nowhere.
04:54
Ten goes to seven, nowhere, nowhere, nowhere, ten goes to seven, nowhere, nowhere, nowhere, seven goes to six, six, six, six, six goes back to one.
05:03
Nowhere, nowhere, one goes to five.
05:07
Five goes to four, nowhere, nowhere, nowhere, nowhere, four goes back to one.
05:12
Okay, so we're done with the cycle.
05:14
Okay, i haven't used two yet.
05:16
Two goes to three.
05:18
Nowhere, nowhere.
05:19
Three goes back to two.
05:21
Nowhere.
05:22
Okay, so two actually just goes to itself, so i don't need to include it.
05:26
Okay, so we did two.
05:28
Just check that off.
05:29
Four, five, six.
05:31
Okay, seven's next.
05:33
Nowhere.
05:34
Seven goes to two.
05:35
Nowhere, nowhere.
05:36
Two goes back to seven.
05:37
Okay, so seven goes to itself.
05:40
Let me do include that either.
05:43
Eight is next...