# How to find the number of orbits

In our lecture notes, i am confused on how to determined the number of orbits with the given permutations in S4. If i consider the permutation (12) . I can also rewrite it as cycle decomposi- tion and that is (12)(3)(4). In practice we simply matters by not writing cycles of length 1 and that makes me understand why (3)(4) is not added in the above. In the example it says that the number of orbits is 3 and the 3 orbits are listed: {{1,2},{3},{4}.I was thinking may be our lectrure counted the number of sub-braces which is 3. Now, if we consider the permutation (123) which can be further written in cycle decomposition: (123)(4). It says the number of orbits is 4 and are listed : {{1,2,3},{4}}. Now, while trying to use my assumption above, i could not apply it in here. The number of sub-braces is 2 but the number of orbits is 4. I am confused!!! For the permutation (1234), The number of orbits is 1 which is listed: {{1,2,3,4}} which agrees with my assumption above. Can anyone show me the proper way of finding the number of orbits?

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Do you understand what an orbit of a group action is? – Montez Jan 29 '13 at 0:24
$(123)(4)\in S_4$ acting on $\{1,2,3,4\}$ has two orbits, $\{1,2,3\}$ and $\{4\}$, not four. (If you were considering $(123)$ as an element of $S_6$, then it would have four orbits.) – anon Jan 29 '13 at 0:27
ohh thankx anon!! that means there is an error in our lecture note!! – Timoci Lagilevu Jan 29 '13 at 0:46
Yes, there is an error in your lecture notes. You are correct that the permutation $(123)$ has two orbits, which is, as you correctly observe, is the number of sub-braced elements in the set of orbits. – amWhy Jan 30 '13 at 23:56