I am looking an automorphism (isomorphism onto itself) of S7 (symmetric group on 7 letters). The extra condition I have been given is that no element is allowed to remain fixed after applying said isomorphism (eg 1 should not be sent back to 1). To make this clearer, I am looking for a function, f, on 7 letters that:
1. Is a bijection
2. Satisfies f(xy)=f(x)f(y)
3. Leaves no element of S7 fixed
I have tried fiddling around with a lot of modular arithmetic and conjugacy classes etc, but it is hard to find such a function. Perhaps such a function does not exist?
Thanks in advance