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?
2026-04-05 23:00:17.1775430017
How to find the number of orbits
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