How to find all elements of S4 that satisfy the equation x^4=e?

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SUMMARY

The discussion focuses on identifying all elements of the symmetric group S4 that satisfy the equation x^4 = e, where e is the identity element. The conclusion reached is that the solutions consist of all cycles of lengths that divide 4, specifically 1, 2, and 4. The final set of solutions includes elements such as e, (1 2), (1 3), (1 4), (2 3), (2 4), (3 4), (1 2 3 4), (4 3 2 1), (3 4 1 2), (1 2)(3 4), (1 3)(2 4), and (1 4)(2 3). The discussion also emphasizes the importance of understanding cycle notation and the distinction between cycles and products of cycles.

PREREQUISITES
  • Understanding of symmetric groups, specifically S4.
  • Knowledge of cycle notation in permutation groups.
  • Familiarity with the concept of order of an element in group theory.
  • Basic understanding of even and odd permutations.
NEXT STEPS
  • Study the properties of symmetric groups, particularly S5 and higher.
  • Learn about the classification of permutations into even and odd.
  • Explore the concept of cycle types and their applications in group theory.
  • Investigate the generalization of the equation x^n = e in symmetric groups.
USEFUL FOR

Mathematicians, particularly those specializing in group theory, students studying abstract algebra, and anyone interested in the properties of symmetric groups and permutation cycles.

AdrianZ
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well, I'm thinking to answer the question in a generalized way. I want to find all the elements of Sn that satisfy the equation xn=e.
well, if m|n and xm=e then xn=e, hence, if we raise x (which is of order m and m divides n) to the exponent n we'll have xn=e. so the answer will be this:
S={all cycles of the length m s.t. m|n}

Is that a true conclusion?

As an example, in S4, since divisors of 4 are 1,2 and 4 the solutions will be all cycles with lengths 1,2 and 4. namely, {e,(1 2),(1 3), (1 4), (2 3), (2 4), (3 4), (1 2 3 4), (4 3 2 1), (1 3)(2 4), (1 2)(3 4), (1 4)(2 3)}
Is there anything I'm missing?
 
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AdrianZ said:
well, I'm thinking to answer the question in a generalized way. I want to find all the elements of Sn that satisfy the equation xn=e.
well, if m|n and xm=e then xn=e, hence, if we raise x (which is of order m and m divides n) to the exponent n we'll have xn=e. so the answer will be this:
S={all cycles of the length m s.t. m|n}

Is that a true conclusion?

As an example, in S4, since divisors of 4 are 1,2 and 4 the solutions will be all cycles with lengths 1,2 and 4. namely, {e,(1 2),(1 3), (1 4), (2 3), (2 4), (3 4), (1 2 3 4), (4 3 2 1), (1 3)(2 4), (1 2)(3 4), (1 4)(2 3)}
Is there anything I'm missing?

First of all something pedantic: I wouldn't call (1 2)(3 4) a cycle. It is rather the product of cycles. So I would say that your set would rather be the cycles generated by the cycles of 1, 2 and 4.

I think the conclusion holds true. To prove it, you should use that if x_i are disjoint cycles of length k_i, then the order of x_1...x_l is lcm(k_1,...,x_l).
 
micromass said:
First of all something pedantic: I wouldn't call (1 2)(3 4) a cycle. It is rather the product of cycles. So I would say that your set would rather be the cycles generated by the cycles of 1, 2 and 4.

even this statement is not true. for example, (1 3)(1 2) = (1 2 3), so we don't want the set generated by cycles of length 1,2 and 4, but rather the set of products of disjoint cycles generated by cycles of length 1,2 and 4.

we can disregard cycles of length 1, as these are all the identity map.
 
Yea. so, the correct statement would be: S={all cycles of Sn generated by disjoint cycles of length m such that m divides n} would be the solutions of the equation xn=e.
if we stop being pedantic, have I found the solutions of that equation correctly or there are more?
 
There are more.
Did you enumerate and check all of them?
 
I like Serena said:
There are more.
Did you enumerate and check all of them?

well, maybe I should include (3 4 1 2) as well?
I didn't enumerate them, but tried to find them mentally. I considered all transpositions, because they are of order 2. They can be found easily, then I considered all cycles of the length 4 with their inverses (there was only one such cycle), and then I added all cycles that were products of disjoint cycles of order 2. I think I missed (3 4 1 2), it's of order 2 and it's its own inverse too.
S={e,(1 2),(1 3),(1 4), (2 3), (2 4), (3 4), (1 2 3 4), (4 3 2 1), (3 4 1 2), (1 2)(3 4),(1 3)(2 4), (1 4)(2 3)}
well, Is there any other solutions to the equation or I'm done?
 
Did you know that (1 2 3 4)=(3 4 1 2)?
How many cycles of length 4 did you find?

Do you know how many elements S4 has?
How many could you eliminate?
Do you have all the elements that you could not eliminate?
 
I like Serena said:
Did you know that (1 2 3 4)=(3 4 1 2)?
Oops, Yea, I'm sorry. I thought (3 4 1 2) meant (1 3)(2 4). my bad.

How many cycles of length 4 did you find?
2.

Do you know how many elements S4 has?
4! = 24.

How many could you eliminate?
Do you have all the elements that you could not eliminate?
all elements of order 3 can be eliminated. right?
well, I have 12 elements. but I guess there must be more. I'm not sure. Could you tell me one element that is not included in my list?
 
You have 12 elements of order 3?
How did you count them?

And 2 elements of order 4?
There should be more.
 
  • #10
I like Serena said:
You have 12 elements of order 3?
How did you count them?
No, I have so far counted 12 solutions. I didn't say I have 12 elements of order 3. If I had 12 elements of order 3 then I was done. lol

And 2 elements of order 4?
There should be more.
That's exactly where I'm stuck.
Is there any theorem that tells us how many elements of a specific order we have in a group?
 
  • #11
Suppose you were to create a 4-cycle from scratch.
How many choices do you have for the 1st number?
And the following numbers?
 
  • #12
I like Serena said:
Suppose you were to create a 4-cycle from scratch.
How many choices do you have for the 1st number?
And the following numbers?

well, I'm not so sure but I will just tell you my guess based on intuition.
I'll have 4 choices for the first letter, but just after I chose the first number, the arrange of the 3 next numbers would be known, I can either shift all the remaining numbers to right or I can shift them to left depending on the first number I've chosen. Is that right? so if that's right, we might have 8 cycles of order 4 at maximum.
 
  • #13
Let's not make assumptions about the other 3 numbers yet.
Let's start with counting how many sequences of 4 different numbers you can make.

On another angle, consider (1 2 3 4).
Which other 4-cycles are identical to this one?
 
  • #14
I like Serena said:
Let's not make assumptions about the other 3 numbers yet.
Let's start with counting how many sequences of 4 different numbers you can make.
What kind of sequences you mean? just to understand the terminology.

On another angle, consider (1 2 3 4).
Which other 4-cycles are identical to this one?
(1 2 3 4),(2 3 4 1), (3 4 1 2), (4 1 2 3) are identical to this one.
 
  • #15
AdrianZ said:
What kind of sequences you mean? just to understand the terminology.

I mean sequences like:
1 2 3 4
1 2 4 3
...

How many are there?


AdrianZ said:
(1 2 3 4),(2 3 4 1), (3 4 1 2), (4 1 2 3) are identical to this one.

Good!
So a given 4-cycle comes in 4 variants.
 
  • #16
I like Serena said:
I mean sequences like:
1 2 3 4
1 2 4 3
...

How many are there?
If you're putting no additional conditions, there are 4!=24 permutations on 4 letters.

Good!
So a given 4-cycle comes in 4 variants.
True. well?
 
  • #17
AdrianZ said:
If you're putting no additional conditions, there are 4!=24 permutations on 4 letters.True. well?

Yes. Both true!

Since there are 24 sequences of 4 letters and since each 4-cycle comes in 4 flavors.
How many different 4-cycles do you think there are?

Could you enumerate them now?
 
  • #18
I like Serena said:
Yes. Both true!

Since there are 24 sequences of 4 letters and since each 4-cycle comes in 4 flavors.
How many different 4-cycles do you think there are?

Could you enumerate them now?

(1 2 3 4), (2 3 4 1), (3 4 1 2), (4 1 2 3)
(1 2 4 3), (2 4 3 1), (4 3 1 2), (3 1 2 4)
(1 3 2 4), (3 2 4 1), (2 4 1 3), (4 1 3 2)
(1 3 4 2), (3 4 2 1), (4 2 1 3), (2 1 3 4)
(1 4 3 2), (4 3 2 1), (3 2 1 4), (2 1 4 3)
(1 4 2 3), (4 2 3 1), (2 3 1 4), (3 1 4 2)

All the permutations in the same row are identical. so I think we'll have 6 different 4-cycles. Am I right?
 
  • #19
Yep! :smile:
 
  • #20
So do you think you can deduce the number of 3-cycles now?
 
  • #21
this is how i count n-cycles in Sn (this works for any n, but i will use n = 4):

since any 4-cycle in S4 involves 1, we may as well start with it:

(1...

i have 3 choices for my next element (the image of 1):

(1 2...
(1 3...
(1 4...

after i make my next choice, i'll have them all:

(1 2 3 4)
(1 2 4 3)
(1 3 2 4)
(1 2 4 2)
(1 4 3 2)
(1 4 2 3)

that makes 6.

in general, in Sn, we'll get (n-1)(n-2)...(2) = (n-1)! possible distinct n-cycles.
 
  • #22
I like Serena said:
So do you think you can deduce the number of 3-cycles now?

Yes. Actually I was thinking about generalizing this idea to other cases as well.
For 3-cycles, each permutation will come in 3 different variants, and we'll have 24 permutations that we can consider, so the answer should be 8 and the desired 3-cycles are as follows:
(1 2 3),(2 3 1),(3 1 2)
(1 3 2),(3 2 1),(2 1 3)
(1 2 4),(2 4 1),(4 1 2)
(1 4 2),(4 2 1),(2 1 4)
(2 3 4),(3 4 2),(4 2 3)
(2 4 3),(4 3 2),(3 2 4)
(3 4 1),(4 1 3),(1 3 4)
(3 1 4),(1 4 3),(4 3 1)

Right?
 
  • #23
AdrianZ said:
Yes. Actually I was thinking about generalizing this idea to other cases as well.
For 3-cycles, each permutation will come in 3 different variants, and we'll have 24 permutations that we can consider, so the answer should be 8 and the desired 3-cycles are as follows:
(1 2 3),(2 3 1),(3 1 2)
(1 3 2),(3 2 1),(2 1 3)
(1 2 4),(2 4 1),(4 1 2)
(1 4 2),(4 2 1),(2 1 4)
(2 3 4),(3 4 2),(4 2 3)
(2 4 3),(4 3 2),(3 2 4)
(3 4 1),(4 1 3),(1 3 4)
(3 1 4),(1 4 3),(4 3 1)

Right?

i prefer to think of how many ways we can choose 3 objects out of 4:

4!/(3!1!) = 4, and for each set {a,b,c}, we can form two different 3-cycles:

(a b c) and (a c b)
 
  • #24
Right!
 
  • #25
Deveno said:
i prefer to think of how many ways we can choose 3 objects out of 4:

4!/(3!1!) = 4, and for each set {a,b,c}, we can form two different 3-cycles:

(a b c) and (a c b)

Yea, That's exactly what I did.
 
  • #26
so, to find the number of k-cycles in Sn, you take n choose k times (k-1)!, which is how many k-cycles you can form in Sk.
 
  • #27
Hey man, I thought group theory sucks, but now I think group theory rocks. everything looks so beautifully consistent. I believe I should solve more problems in group theory rather than just dealing with the concepts abstractly.

I'll try to find how many solutions the equation x5=e can have in S5. then I'll try to guess how many solutions the equation xn=e can have in Sn and will write down my thoughts here. it looks to be a good food for thought. Thank you guys for your helps, especially I like Serena.

One more question, Is it always possible to solve an equation like axn=b in Sn? When it's possible?
 
  • #28
AdrianZ said:
Hey man, I thought group theory sucks, but now I think group theory rocks. everything looks so beautifully consistent. I believe I should solve more problems in group theory rather than just dealing with the concepts abstractly.

I'll try to find how many solutions the equation x5=e can have in S5. then I'll try to guess how many solutions the equation xn=e can have in Sn and will write down my thoughts here. it looks to be a good food for thought. Thank you guys for your helps, especially I like Serena.

Looks like a good plan!


AdrianZ said:
One more question, Is it always possible to solve an equation like axn=b in Sn? When it's possible?

No, it's not always possible.
It depends on the order of the permutations involved, and it also depends on whether the permutations are even or odd.
 
  • #29
I like Serena said:
No, it's not always possible.
It depends on the order of the permutations involved, and it also depends on whether the permutations are even or odd.

Would you explain more please?

We found out that there are 1 one-cycle, 6 different 2-cycles, 8 different 3-cycles and 6 different 4-cycles in S4. but if we add 1+6+8+6 it'd be equal to 21, not 24. How so?
 
  • #30
AdrianZ said:
Would you explain more please?

We found out that there are 1 one-cycle, 6 different 2-cycles, 8 different 3-cycles and 6 different 4-cycles in S4. but if we add 1+6+8+6 it'd be equal to 21, not 24. How so?

Because there are three elements we missed: (1 2)(3 4) is one of them. Can you find the others?
 

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