Do Permutations Commute Only When Identical?

[C_Ovidiu]

Homework Statement

let there be a and b (b taking any value) two permutation with the same grade .demonstrate that
ab=ba <=> a=e(e=the identical permutation) .

Homework Equations

e=(1234)
(1234)

The Attempt at a Solution

Don't have a clue with what to start/end

 

[EnumaElish]

How do you define grade?

Which operation does ab indicate?

 

[C_Ovidiu]

Say you have b= 3 4 2 1 the grade is 4

ab means a*b that means ab=(a(b(1)), a(b(2)), a(b(3))... a(b(n)) )

 

[EnumaElish]

Can you give an example with specific a and b, and derive ab?

 

[C_Ovidiu]

a= 1 3 4 2
b= 2 4 1 3
ab= 3 2 1 4

 

[EnumaElish]

Can you explain why a(b(1)) = 3?

 

[C_Ovidiu]

b(1)=2 , a(2)=3 ===>a(b(1))=3

 

[EnumaElish]

I'd start with:

Let
a= a1 a2 a3 a4
and let
b = b1 b2 b3 b4.

Then
ab = a(b(1)) a(b(2)) a(b(3)) a(b(4))
and
ba = b(a(1)) b(a(2)) b(a(3)) b(a(4))

Therefore ab = ba implies
a(b(1)) = b(a(1)) , ..., a(b(4)) = b(a(4)).

What are the properties of permutations?