Question: I had a question regarding the proof of #46 in Section 8. This is the one that we have to show that S of n > or = to 3 is nonabelian. My first instincts are to split it into 2 cases. Equal to 3 and greater than 3. Will this be a good idea? Is it sufficient to give a couple of permutations and show that it is not commutative? or would you want us to list every single permutation?

Answer: Actually, you can treat those two cases the same.  Here’s a hint:

Define f from {1,2,3,…,n} to {1,2,3,…,n} by

f(1)=2

f(2)=1

f(x)=x if x>2

Then f is in S_n.

Now find g in S_n such that fg is not equal to gf.