S(n) is not abelian

1. Apr 12, 2005

TsunamiJoe

I'm having troubles with this problem here -

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

now right now, im 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.

2. Apr 12, 2005

Hurkyl

Staff Emeritus
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.

3. Apr 13, 2005

Don Aman

eh. Once you've done S3, you're done, since Sn contains S3 as a subgroup.

4. Apr 13, 2005

Hurkyl

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

5. Apr 13, 2005

TsunamiJoe

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

6. Apr 13, 2005

Hurkyl

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

7. Apr 18, 2005

TsunamiJoe

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