Understanding Symmetric Groups: S4 Order & Products

In summary, the order of S4, the symmetric group on 4 elements, is the number of permutations on 4 elements. The process for computing products in order of S4 involves thinking of cycles as functions and using composition to simplify them into disjoint cycles. The correct answers for the given products in order are (1342), (2431), and (13)(24).
  • #1
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What is the order of S4, the symmetric group on 4 elements? Compute these products in order of S4: [3124] o [3214], [4321] o [3124], [1432] o [1432].

Can I get help on how to do this. The solution's manual gives the answer on how to do the last two, but I don't understand the process whatsoever. Any help?
 
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  • #2
Anybody know a good process of doing this?
 
  • #3
For the order you count - it's the number of permutations on 4 elements.

You need to think of a cycle as a function: it takes an element of 1,..,4 (or n rather than 4 in general), and returns another element. If you give it x, then it either returns the element to the right of x as listed in the cycle, wrapping round from beginning to end, if x is in the cycle, or x if x is not in the cycle.

Thus (123) we think of as the function that sends 1 to 2, 2 to 3, 3 to 1, and leaves 4 alone.

Composition of cycles is just composition of these permutation functions, remembering functions 'start on the right' i.e. fg means do g first, then f. To simplify some composition to disjoint cycles, then you just need to see what happens to all of the possible inputs. For example

(123)(132)

we only need to see what happens to 1,2, and 3; 4 is unmoved.

1 is sent to 3 by (132), so I need to see where (123) sends 3. It sends it to 1.

2 is sent to 1 by (132), and 1 is sent to 2 by 123.

3 is sent to 2 by (132) and then 2 is sent to 3 by (123).

This (123)(132) fixes everything, and is the identity.
 
  • #4
Thanks Matt for the reply.

To check if I understand correctly:
[3142]o[3214]

1 is sent to 4 by (3214), so I need to see where (3142) sends 4. It sends it to 2.

2 is sent to 1 by (3214), and 1 is sent to 4 by 3142.

3 is sent to 2 by (3214) and then 2 is sent to 3 by (3142).

4 is sent to 3 by (3214) and then 3 is sent to 1 by (3142).

ANSWERS:

(1342),(2431),(1234)
Is this correct?
 
  • #5
You have the right idea (it would help if you wrote out what you think (3142)(3214) is, though), but the wrong answers. For instance, the third one is (13)(24); do you see why?
 
  • #6
Aww yes, that makes sense. Thanks for the help!
 

What is a symmetric group?

A symmetric group is a mathematical group composed of all possible permutations of a given set. In other words, it is a group that contains all the possible ways to rearrange the elements of a set.

What is the order of S4?

The order of S4, also known as the symmetric group of degree 4, is 24. This means that there are 24 possible permutations of a set with 4 elements.

How do you calculate the order of S4?

The order of S4 can be calculated by using the formula n!, where n is the number of elements in the set. In this case, n=4, so the order of S4 is 4!=4x3x2x1=24.

What is the product of two elements in S4?

The product of two elements in S4 is found by performing the first permutation, followed by the second permutation. For example, if we have (123) and (12), the product would be (13).

What is the identity element in S4?

The identity element in S4 is the permutation that leaves all the elements unchanged. In this case, it would be (1)(2)(3)(4), which is equivalent to (1234).

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