# Proof: S_n is Not Abelian for n >= 3

• TsunamiJoe
In summary, the conversation discusses the problem of proving that S_N is not abelian for any n >= 3. The individuals in the conversation suggest experimenting with S_3 and S_4 to find a pattern and also mention the importance of understanding the elements in S_N and the concept of abelian. One person mentions needing their textbook for other exercises, but it is clarified that it is not necessary for this problem.
TsunamiJoe
I'm having troubles with this problem here -

Show $$S_N$$ is not abelian for any $$n >= 3$$

now right now, I am simply lost, of course its late at night so that might be why, so if some help could be provided that would be appreciated, also i would like it if you didnt simply give the proof, but also explained it.

Experiment. Start with S_3 and try to find two permutations that don't commute. Then, try S_4. Look for a pattern you can exploit.

Hurkyl said:
Experiment. Start with S_3 and try to find two permutations that don't commute. Then, try S_4. Look for a pattern you can exploit.
eh. Once you've done S3, you're done, since Sn contains S3 as a subgroup.

Yep. I'm hoping TsunamiJoe will notice that his example for S_3 will work for S_4 and all the others.

|~|will respond with answer tomarow, was stupid and left textbook at school|~|

Do you need your textbook for this? You just need to know what kind of elements are in S_N and what abelian means.

|~| that was not the only exercise to work on, and i try to keep a train of thought when doing maths like these |~|

## 1. What is S_n?

S_n is the symmetric group of degree n, which is the group of all permutations of n distinct objects. In other words, it is the group of all possible ways to arrange n objects in a line.

## 2. What does it mean for a group to be abelian?

A group is considered abelian if its group operation is commutative, meaning that the order in which elements are multiplied does not affect the result. In other words, if a and b are elements of an abelian group, then a * b = b * a.

## 3. Why is it important to prove that S_n is not abelian for n >= 3?

Proving that S_n is not abelian for n >= 3 is important because it helps us understand the structure of this group and its properties. It also allows us to make connections to other areas of mathematics and to apply this knowledge in various fields, such as computer science and physics.

## 4. How is the proof that S_n is not abelian for n >= 3 typically done?

The proof typically involves constructing specific permutations and showing that they do not commute with each other, thus demonstrating that the group operation is not commutative. This can be done using various techniques, such as showing that the order of the elements in the group is different when multiplied in different orders.

## 5. Are there any exceptions to the fact that S_n is not abelian for n >= 3?

Yes, there are two exceptions: S_1 and S_2. These are the only cases where S_n is abelian, as they only have one and two distinct elements respectively, making it impossible for the group operation to be non-commutative.

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