The discussion revolves around the symmetric group S_5 and the permutations of specific subsets, particularly focusing on the elements {2,5} and {3}. Participants explore the nature of identity permutations and the representation of cycles in permutations.
The conversation is ongoing, with participants clarifying concepts related to permutations and exploring the implications of representing identity permutations. Some guidance on the number of permutations based on the number of elements has been provided.
Participants are considering the conventions of writing permutations, particularly regarding 1-cycles and identity elements, as well as the factorial representation of permutations based on the number of elements.
Ted123 said:Say we have the symmetric group S_5.
The permutations of \{2,5\} are the identity e and the transposition (25).
But what are all the permutations of \{3\}? Is it e and the 1-cycle (3)?
Dick said:What do you think would be the difference between e and (3)? Aren't they both the identity permutation?
Ted123 said:Oh yeah of course. You don't have to write 1-cycles in a permutation so e=(1)(2)(3)(4)(5)=(1)=(2)=(3)=(4)=(5)=(1)(2)=(1)(3) etc.