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Let $S_4$ act on itself by conjugation. How many orbits are there for this action? How many elemnts in each orbit? [HINT: The orbits are 1 - 1 correspondance with the possible types of cycle decomposition.]

From the hint, I get that the different types of cycle decompositions are:

$S_4 = (1234)$

  • (id)
  • (12)(34)
  • (123)(4)
  • (1234)

How do I calculate the number of elements in these orbits then?

EDIT: Missed out (12)(3)(4)

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    If worst comes to worst, you could just write out all the elements of $S_4$ --- there are only $24$ of them --- and count how many are $4$-cycles, how many are $3$-cycles, how many are products of two transpositions, etc. Not the high road to the answer, but you might learn something about that group that way.2012-12-12

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