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**TeaWithoutMussolini** 1) "Explain what is meant by $\displaystyle \pi$ is a permutation of $\displaystyle \bar{n} =(1,2...n)$. The permutation $\displaystyle \pi$ is 123456 -> 635124, write in cycle notation. Find $\displaystyle {\pi}^{-1}$. How many permutations of $\displaystyle {S}_{6 }$ are there in the conjugacy class of $\displaystyle \pi$"

Ok for the first part I don't know how to word it. I know what a permutation does, but I don't know how to call it. $\displaystyle \pi$ sends each element to another? I don't know how to define this. Cycle notation I make it (1 6 4)(2 3 5). Just need a confirmation on this one. Pi^-1, = 452631. The conjugacy classes part I have no idea. I think it means with the same cycle types? But I don't know how to do this?

Also, how many permutations of 6 are there in total? Is it 6!?

2) "The following are permutations of 9. a=(12)(345)(78), b=(1234)(6789), c=(197)(34)(58). Calculate bc. Calculate the conjugate c.b.c^-1.

Does there exist a permutaiton oao^-1 = b, or oao^-1 = c."

Now I remember a rule which says the conjugate sends the part of one to the other, the corresponding part. So a conjugate of something has to have the same cycle type? So for this there would be one for c but NOT a? And for oao^-1 = c, o must be... I forget how to do it?