Confusion on permutation from lecture notes

Question

This lecture notes on page 1 gives permutation

image1

Then asks this question: "Carrying out the “shuffles” described by P , what will be the new arrangement of the numbers {1,2,3,4} ?"

Then on page two gives an answer:

image2

From all the notation I had seen before I thought the answer had to be: {1,2,3,4}->{2,3,4,1} (after applying permutation from first image).

The author even goes to say that {2,3,4,1} is not the answer and students confuse it.

Can someone please clarify is it something with me or lecture notes is wrong or using different notation?

e.g. In other notes, here, after definition 3.1 (page 5) author gives a permutation and says "By the permutation σ, a is sent to b, b is sent to c, and c is sent to a". If I had applied similar reasoning and notation to the question I asked originally from first lecture notes, and applied sigma from latter lecture notes, to {1,2,3,4} answer had to be {2,3,4,1}, isn't it?

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I found a related question: Convention on Cauchy's two line notation for permutations, though there isn't a very high voted answer.

Answer 1

Did you read the explanation in Section 6.3? I don't really see how it can be made any plainer. Look at the pictures again. If you still can't see it, please try to indicate exactly where in the discussion you get lost.

I remember this confused me too, when I first saw it, 50 years or so ago. You have to realize that we permute things other than numbers. While it's very tempting to look at the notation and think, "That means 1 maps to 2, 2 maps to 3, and so on," that wouldn't make any sense if were permuting triangles, say or sausages. What it means is "Whatever is in position 1 gets moved to position 2, whatever is in position 2 gets moved to position 3, etc." Look at the explanation with the boxes and marbles again.

EDIT It appears that are two conflicting two-line notations for permutations, according to the Wikipedia article. If I was ever aware of this, I've forgotten it. There's a more compact notation called the cycle notation, that is generally used, so you don't see the two-line notation much. What I'll call the "alternate notation" is great for directly describing a rearrangement of the numbers, but it's not clear to me what it would mean to perform it a second time. I looked through the entire chapter to see if I could find a concrete example of composition of permutations with the alternate definition, and I couldn't. The first definition makes more sense to me if one is thinking of permutations as bijections of a set to itself, and the second makes more sense if one is thinking of a permuation simply as a rearrangement.

Answer 2

I will suggest you just forget about that shuffle example.It is actually confusing and will make you more irritate.But I will try some to make things clear. Author haven't ask actually in that shuffle example to show what is going where as you are saying 1 is going to 2 and same for other numbers. What he is asking is inverse of the permutation that I think you haven't studied yet. I think you also don't know how to write permutations in cyclic form because $(1,2,3,4)$ is same as writing $(2,3,4,1)$..I don't want to make you more confused but suggest just to skip that part.. Go ahead and then come back..or you can use different reference book (Contemporary Abstract Algebra by Joseph Gallian..believe me this is the best book for beginners.)