How many elements of order 2 are contained in S_4?

[Samuelb88]

Homework Statement

How many elements of order 2 does the symmetric group [itex]S_4[/itex] contain?

Homework Equations

The Attempt at a Solution

I know that transpositions have order two. Moreover, any k-cycle has order k. Thus there are six elements with order two contained in [itex]S_4[/itex]?

Is this all? Seems too simple.

 

[ehild]

How many elements are there in S4 and what are they?

ehild

 

[Samuelb88]

ehild,

There are 4! = 24 elements in [itex]S_4[/itex]. The elements are all such permutations of {1,2,3,4}.

Here is how I see the problem:

Let [itex]\sigma \in S_4[/itex]. Then either:
1. [itex]\sigma = (a_1 a_2)[/itex] and [itex](a_1 a_2)^2 = e[/itex] [itex]\Rightarrow[/itex] order 2.
2. [itex]\sigma = (a_1 a_2 a_3[/itex] and [itex](a_1 a_2 a_3)^3 =e[/itex] [itex]\Rightarrow[/itex] order 3.
3. [itex]\sigma = (a_1 a_2 a_3 a_4)[/itex] and [itex](a_1 a_2 a_3 a_4)^4 = e[/itex] [itex]\Rightarrow[/itex] order 4.

Thus there are 6 permutations of order 2 in [itex]S_4[/itex]. Is this not correct?

 

[ehild]

Oh, you mean the permutation group? I thought it was the point group called S4 - sorry.

ehild

 

[I like Serena]

Thus there are 6 permutations of order 2 in [itex]S_4[/itex]. Is this not correct?

No, this is not correct.
Did you find how many of order 3 and 4 there are?
Do they add up to 24?

 

[ehild]

I found a useful page http://groupprops.subwiki.org/wiki/Element_structure_of_symmetric_group:S4

ehild

 

[AdrianZ]

well, I got an idea but am not sure if I'm right because I haven't had abstract algebra yet. well, what you're claiming is that in a cyclic group G an element in G is a transposition if and only if It is of order 2. well, It's obvious that any transposition element in G is of order 2 but can we say that any element of order 2 is a transposition?
if yes, then your question would become that in how many ways we can permute two letters from n letters keeping the others the same position they are. That would be an easy problem in combinatorics and discrete math.

 

[micromass]

Hi Samuelb88!

You are certainly correct that there are 6 transpositions in S₄, i.e. there are 6 elements of the form (a b). These are

[tex](1~2),(1~3),(1~4),(2~3),(2~4),(3~4)[/tex]

However, these are not the only elements of order 2! For example

[tex](1~2)(3~4)[/tex]

is also of order 2, so you got to count this one too!

 

[Samuelb88]

Oh right! So that means there are six transpositions and three disjoint transpositions in [itex]S_4[\itex].

 

[micromass]

Yes, that sounds right!

 

[Samuelb88]

Thanks guys!