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I cant seem to find anything on the internet on this subject , and the professor did not explain it too well, in short the following is unclear to me how is $$(1 3 4)(236)=(24136)$$and $$(123)(435)(1346)=(154623)$$ if someone can explain the rule, that would be great, in the comments.

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(1 3 4)(2 3 6)=(2 4 1 3 6)

I'd do each element in order. The important thing (for me) to remember is to work from right to left (on the lhs). You can start with any element, but starting with 1, (2 3 6) which is applied first, does not change 1, so you proceed to (1 3 4) and see that 1 is sent to 3. So if you were building the rhs you would now have (...1 3...). Now you want to know what happens to 3. In (2 3 6) you see that 3 is sent to 6. But you can't stop here without checking if (1 3 4) changes 6. It doesn't, so you now have (...1 3 6...). Now what happens to 6? (2 3 6) sends 6 to 2 (remember that these are cycles and they wrap around). Again (1 3 4) leaves 2 alone so you have (...1 3 6 2...). Next you see that 2 is sent to 3 by (2 3 6). This time when you check if (1 3 4) affects 3, you see that it does: It sends it to 4. So the original element 2 is sent to 4 (via 3). Thus you now have (...1 3 6 2 4...). Lastly 4 is sent to 1 so the cycle is complete: You can write (1 3 6 2 4). Finally, you have to note that a given cycle can be written starting with any element (again remembering the wrapping around mentioned above). So (1 3 6 2 4) represents the same cycle as (2 4 1 3 6).

I think this is enough information to do the second example? Just follow the exact same procedure. The main difference is that because there are 3 cycles on the lhs, you have to check (again reading rtl) each of the 3 in turn (just as was done for the 2 consecutive that have to be checked in the first example).

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Start with 1: $(236)$ takes $1$ to $1$, and than $(134)$ takes $1$ to $3$. So write down

$$(13$$

continue with $3$. $(236)$ takes $3$ to $6$, and than $(134)$ takes $6$ to $6$, so add:

$$(136$$

Continue with $6$, $(236)$ takes $6$ to $2$, and $(134)$ takes $2$ to $2$, so add:

$$(1362$$

Now I think you see how to continue from here. Just remember to start from the right most permutation.

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