Permutations of a single number in the symmetric group

[Ted123]

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]

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 $(3)$?

What do you think would be the difference between e and (3)? Aren't they both the identity permutation?

 

[Ted123]

What do you think would be the difference between e and (3)? Aren't they both the identity permutation?

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.

 

[Dick]

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.

Right. The number of permutations on n elements is n!. So for 2 elements you have two permutations, for 1 element you have one.