Understanding Symmetric Groups: S4 Order & Products

[needhelp83]

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?

 

[needhelp83]

Anybody know a good process of doing this?

 

[matt grime]

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.

 

[needhelp83]

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?

 

[matt grime]

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?

 

[needhelp83]

Aww yes, that makes sense. Thanks for the help!