13. Consider the permutations ? = (1 5 3 2)(4 6) and ? = (2...

13. Consider the permutations ? = (1 5 3 2)(4 6) and ? = (2 4 6 5)(1 3) of {1, 2, 3, 4, 5, 6} expressed in cycle notation. Which one of the following is a correct expression for a permutation ? with the property ? = ??¹?? where we compose from left to right? (a) ? = (1 4 5 2)(3 6) (b) ? = (1 2 4 5)(3 6) (c) ? = (1 4 6 2 3) (d) ? = (1 6 3 4)(2 5) (e) ? = (1 4 3 5 6) 14. Consider the permutations ? = (5 2 1 4 3), ? = (1 3)(2 4 6), ? = (1 2 4)(3 5 6) of {1, 2, 3, 4, 5, 6} expressed in cycle notation. Which one of the following is correct? (a) ? and ? are odd, and ? is even. (b) ? and ? are even, and ? is odd. (c) ? and ? are even, and ? is odd. (d) ? and ? are odd, and ? is even. (e) ? and ? are even, and ? is odd.

Transcript

00:01 For this problem, we are given alpha and gamma, and we are asked to find which beta satisfies gamma equals beta inverse alpha beta.

00:11 So first off the bat, we want something to equal beta, or we want something to equal gamma.

00:19 And gamma has a nice little two cycle here that would be really easy to check, just off the bat.

00:25 So we're going to go through all of the betas and pretty much.

00:30 See where beta inverse alpha beta, where one will get sent via each of these betas.

00:39 So just a quick note when we're doing beta inverse, we are pretty much switching the order.

00:50 So 1, 4, 5, 2 goes to 2, 5, 4, 1.

00:55 2 cycles stay the same because when you switch them, it's the same.

01:00 So i want to look at where one is getting sent to two, then two in our alpha is getting sent to one.

01:17 One in beta gets sent to four, so that's immediately a no -go.

01:23 And so we can just automatically cross out a because one does not get sent to three.

01:29 Now we're going to do the exact same process down here just to help us visualize.

01:35 Let's write out what beta should be.

01:37 So it goes 5, 4, 2, 1.

01:40 Like i said before, 2 cycles stay the same.

01:43 We're still interested in where 1 will get sent via this composition.

01:49 So 1 in beta is going to beta inverse is getting sent to 5.

01:53 Then we're going to alpha.

01:55 So 5 and alpha gets sent to 3.

01:58 3 in beta gets sent.

02:00 Sent to six again this is a no -go because one does not go back to three so now we're left with c d and e so again beta inverse that's going to be three two six four one and now we're wanting to see one gets sent to three three gets sent to two and two gets sent to three which is very good because now we have one getting sent to three.

02:36 Now we need to check and make sure that three will get sent back to one at the end of this composition and finish this two cycle for us.

02:45 So now we're going to start with three in beta, inverse, three gets sent to two, and alpha, two gets sent to one, and one gets sent to four.

02:58 So this would actually keep going.

03:01 We really don't care how long because we just know it is not what we want.

03:07 And we can just immediately cross it off.

03:11 Now, let's check d and e.

03:16 So we have beta inverse.

03:19 It's equal to 4, 3, 6, 1.

03:22 Two cycles stay the same when we're inversing.

03:25 Let's see where 1 gets sent.

03:27 One goes to 4, and alpha 4 gets sent to 6.

03:33 Beta 6 gets sent to 3.

03:35 So we are having a good success at this so far.

03:39 So now we need to see where three goes in beta inverse, gets sent to six.

03:44 Six gets sent to four in alpha.

03:47 Then four goes all the way back to one.

03:50 So we have completed a two cycle.

03:52 So this is really, really good.

03:55 However, we can't really stop here because stuff could get messed up in this other part of gamma.

04:05 So let's just go ahead and check e really fast to see if one gets sent to three and that's the end of it.

04:16 Because if one does not get sent to three as a two cycle and e, then we can just automatically say that d is it because the rest of them don't work.

04:27 So beta inverse, for e is going to be six, five, three, four, and one.

04:33 Again, we're interested to see where one's going to end up.

04:37 So one goes to six.

04:39 Oops.

04:42 Then six goes to four and four goes to three.

04:48 So we are kind of off to a good start here.

04:51 Now let's see where three goes and beta inverse goes to four.

04:55 Four goes to six.

04:57 And six goes all the way back to one.

05:00 So now we have two contestants.

05:03 We have d and e because they both have the two cycle of one, three, and them.

05:09 Oops.

05:13 And so now let's go ahead and check, let's go back to d and see where does two get sent...

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