Proof: S_n is Not Abelian for n >= 3

[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.

 

[Hurkyl]

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.

 

[Don Aman]

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.

 

[Hurkyl]

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

 

[TsunamiJoe]

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

 

[Hurkyl]

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

 

[TsunamiJoe]

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