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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)

Kaish
  • 6,126
  • Possible duplicate (at least very similar):http://math.stackexchange.com/questions/251333/orders-of-a-symmetric-group/251350#251350 – John Martin Dec 12 '12 at 22:04
  • @John: You linked to an answer, not a question. There appears to be a widespread misunderstanding that whether a question is a duplicate depends on whether it's been answered before. A question is a duplicate if a sufficiently similar question has already been asked. I do agree that the question to whose answer you linked is sufficiently similar, but it's a borderline case since the formulations of the questions are quite different and it already requires some insight into the problems to recognize the connection. – joriki Dec 12 '12 at 22:13
  • I added another answer to that question, deriving the result in more generality so it can more readily be applied to the present case (and potential future cases). – joriki Dec 12 '12 at 22:34
  • 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. – Gerry Myerson Dec 12 '12 at 23:44

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