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Could someone show me how to multiply in say $S_4$. I know how to multiply say $(4321)(2341)$ but when it comes to ones that do not contain $4$ terms, like $(34)(231)$, I have no idea how to handle this. We do from left to right.

snowman
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4 Answers4

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Don't go from left to right. Go from right to left.

For $(34)(231)$, start with the right pair of parentheses. Pick a number from between that pair and write it in a new pair of parentheses. I will first choose $2$. So I write:

$(2)$

Now, where does $2$ go? It goes to $3$ in the right pair, but then if you work your way to the left pair of parentheses we see in that pair that $3$ goes to $4$. So, ultimately, $2$ goes to $4$. So in your new pair of parentheses, I would write $4$ after $2$.

$(24)$

Now, we need to see where $4$ goes. So start again with the right most pair of parentheses. Since $4$ is not present in there, it means $4$ goes to itself under that pair. Then move to the left, and see where $4$ goes. $4$ goes to $3$, so ultimately, $4$ goes to $3$. So I would write $3$ after $4$.

$(243)$

Now, where does $3$ go? Well, in the right most pair of parentheses, $3$ goes to $1$, and in the left pair, $1$ isn't present, so $1$ goes to $1$ in that pair. So, ultimately, $3$ goes to $1$, so I will write $1$ next to the $3$.

$(2431)$

Now where does $1$ go? In the right pair, $1$ goes to $2$, and in the left pair $2$ is not present, so $2$ goes to itself there. So ultimately, $1$ goes to $2$. So I would need to write $2$ after $1$, but notice that in what I've written, $1$ already goes to $2$, so I don't need to change anything.

So we have $(2431)$. Since every distinct integer in $(34)(231)$ is used, we are done. If any where not used, I would start a new pair to the right of the old pair, like $(2431)()$ and pick one of the unused integers, and go from there.

layman
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    I'm trying to do (13)(234) by your explanation. so start with the 2nd bracket. 2 goes to 3, 3 goes to 1 so 2 goes to 1, so (21....) so far. 1 is not in the second bracket so 1 to 1, then 1 to 3 so 1 goes to 3, so (213..). 3 goes to 4, 4 not appeared so 4 to 4 so 3 to 4. So (2134) but this is wrong according to WFA... – snowman Dec 12 '15 at 23:37
  • @snowman What is WFA? – layman Dec 12 '15 at 23:42
  • wolfram alpha... – snowman Dec 12 '15 at 23:43
  • @snowman What does wolfram alpha say the answer is? – layman Dec 12 '15 at 23:44
  • (1423) which is different to mine... – snowman Dec 12 '15 at 23:48
  • @snowman I'm not sure how you are entering it in Wolfram Alpha -- I don't know how to enter it in there myself. But the answer you posted from there is consistent with using my method only working left to right instead of right to left... But usually we work right to left, so I'm not sure why wolfram alpha is giving you the answer for left to right. – layman Dec 12 '15 at 23:49
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    ok I think I understand. I didn't realise that wolfram was doing it the other way. – snowman Dec 12 '15 at 23:52
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Let me explain this in your given example. First, the product $(3,4)(2,3,1)$ is composition $(3,4) \circ (2,3,1)$. From the composition perspective, let's write $f = (3,4)$ and $g = (2,3,1)$. In this case, we have

$$ \begin{array}{c|c|c|c} x & g(x) & f\circ g(x) & \text{conclusion}\\ \hline 1 & g(1) = 2 & f(2) = 2 & 1 \to 2\\ 2 & g(2) = 3 & f(3) = 4 & 2 \to 4\\ 3 & g(3) = 1 & f(1) = 1 & 3 \to 1\\ 4 & g(4) = 4 & f(4) = 3 & 4 \to 3 \end{array} $$

Using the "conclusion" column, we see that $(3,4)(2,3,1) = (1,2,4,3)$.

Tom
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  • Are you going from right to left? We were told to go left to right... it would be confusing for me if its the opposite – snowman Dec 12 '15 at 22:41
  • @snowman It has to be the opposite. This multiplication of $(34)$ and $(231)$ is actually compositions of functions. If $f$ and $g$ are two functions, then $f \circ g (x)$ is the composition. You do this from right to left. You do $g$ first and then $f$. That's how composition of functions works, and that's exactly what's going on here. – layman Dec 12 '15 at 22:42
  • @user46944 No, it does not have to be one direction or the other. If you write functions to the right of the argument, as in $xf$ or $x^f$ instead of on the left as $f(x)$, then left to right becomes very natural: $x(f \circ g) = (xf)g$. A lot of computer algebra software does this (e.g. GAP, Sage), as do some older textbooks e.g. Herstein. – Ted Dec 16 '15 at 07:40
  • @snowman You should figure out which direction your textbook/instructor wants you to do. The basic process is the same either way. – Ted Dec 16 '15 at 07:44
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$(34)(231)$ means: $(1,2,3,4)\to (2,3,1,4) \to (2,4,1,3)$

Martín Vacas Vignolo
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The usual function notation means fog = f(g).. f acts on g. But (older) group theorists say the "multiplication" (f)(g) of two permutations means g(f). Of course fog is not equal to (f)(g) = gof. The "cycle" notation (34) means 3 -> 4; and 4-> 3 ..all other entries remain unchanged. (231) means 2-> 3; 3 -> 1 ; 1 -> 2. That is (1234) -> (2314). In usual function notation (34)(231) = (1243) o (2314) = (2341) In group theorist notation (34)(231) = (2314)(1243) = (2413).

c.sidney
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