The discussion focuses on the permutations within the symmetric group S_5, specifically examining the permutations of the set {2,5} and the single element {3}. It is established that the permutations of {2,5} include the identity permutation e and the transposition (25). For the single element {3}, the only permutations are the identity e and the 1-cycle (3), which is equivalent to e in notation but not in function. The total number of permutations for n elements is confirmed to be n!, illustrating that for 2 elements there are 2 permutations and for 1 element there is 1 permutation.
PREREQUISITESMathematicians, students of abstract algebra, and anyone interested in group theory and combinatorial mathematics will benefit from this discussion.
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.