# Thread: Abstract Algebra, permutation groups

1. ## Abstract Algebra, permutation groups

I dont even know where to start on this one.

Show for n>2, the group of permutations on n elements is non-abelian.

Thanks!

2. Originally Posted by ziggychick
I dont even know where to start on this one.

Show for n>2, the group of permutations on n elements is non-abelian.

Thanks!
$(1 \ 2)(2 \ 3)=(1 \ 2 \ 3) \neq (1 \ 3 \ 2)=(2 \ 3)(1 \ 2).$ now answer this simple question:

imagine for a second that all symmetric groups were abelian, then is it true that every group would be abelian? (how cool would that be! )

3. Ok. Since every group is isomorphic to some permutation group, if all the permutation groups were abelian, than every group would also be abelian. (That would be super cool. Peace out matrices)

I think I know what to do now. Thanks so much!

4. Originally Posted by NonCommAlg
imagine for a second that all symmetric groups were abelian, then is it true that every group would be abelian? (how cool would that be! )
It would mean all of group theory can be summarized in 60 pages.

Show for n>2
I just want to add, since NonCommAlg was implicit about this, this condition is necessary to have a third element. If n>=2 then writing (12)(23) is not allowed since there is no third element. That is why this condition is necessary.