Permutations of a single number in the symmetric group

In summary, the symmetric group S_5 has permutations of \{2,5\} as the identity e and the transposition (25). There is a difference between e and (3), but they both represent the identity permutation. The number of permutations on n elements is n!, so for 2 elements there are two permutations and for 1 element there is one.
  • #1
Ted123
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0
Say we have the symmetric group [itex]S_5[/itex].

The permutations of [itex]\{2,5\}[/itex] are the identity [itex]e[/itex] and the transposition [itex](25)[/itex].

But what are all the permutations of [itex]\{3\}[/itex]? Is it [itex]e[/itex] and the 1-cycle [itex](3)[/itex]?
 
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  • #2
Ted123 said:
Say we have the symmetric group [itex]S_5[/itex].

The permutations of [itex]\{2,5\}[/itex] are the identity [itex]e[/itex] and the transposition [itex](25)[/itex].

But what are all the permutations of [itex]\{3\}[/itex]? Is it [itex]e[/itex] and the 1-cycle [itex](3)[/itex]?

What do you think would be the difference between e and (3)? Aren't they both the identity permutation?
 
  • #3
Dick said:
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 [itex]e=(1)(2)(3)(4)(5)=(1)=(2)=(3)=(4)=(5)=(1)(2)=(1)(3)[/itex] etc.
 
  • #4
Ted123 said:
Oh yeah of course. You don't have to write 1-cycles in a permutation so [itex]e=(1)(2)(3)(4)(5)=(1)=(2)=(3)=(4)=(5)=(1)(2)=(1)(3)[/itex] 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.
 

1. What is a permutation in the symmetric group?

A permutation in the symmetric group is a rearrangement of a set of objects in a specific order. This order is important because it determines the structure and properties of the group.

2. How many permutations are there of a single number in the symmetric group?

For a single number, there is only one possible permutation in the symmetric group. This is because the number itself is unchanged when rearranged in any order.

3. How is a permutation represented in the symmetric group?

A permutation in the symmetric group is typically represented using cycle notation. This notation shows the movement of each element in the permutation, starting from its original position to its new position.

4. What is the identity permutation in the symmetric group?

The identity permutation is the permutation where all elements remain in their original positions. It is denoted by the symbol "e" and is the neutral element in the symmetric group.

5. How do you calculate the order of a permutation in the symmetric group?

The order of a permutation in the symmetric group is equal to the number of elements in the group. This can be calculated by counting the number of elements in the permutation or by using the formula n!, where n is the number of elements in the group.

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