Question

We have `sigma=((1,2,3,4),(3,4,2,1))inS_4`. How to solve this equation: `x*sigma=sigma*x,x inS_4`?

Answer 1

The only solutions are `sigma^n` for `n=0,1,2,3`

Explanation:

`sigma` will commute with any element of the subgroup generated by `sigma`.

So all of the following are solutions:

`sigma^0 = ((1,2,3,4),(1,2,3,4))`

`sigma^1 = ((1,2,3,4),(3,4,2,1))`

`sigma^2 = ((1,2,3,4),(2,1,4,3))`

`sigma^3 = ((1,2,3,4),(4,3,1,2))`

No other element commutes with `sigma`.