abstract algebra?

Question

i need help getting started on this proof:

Determine all positive integers n such that the mapping α(x)=x^2 from U(n) to U(n) is an automorphism.

Answer 1

Remember that if you can prove that the Hermitian transpose of α(x) is equal to the inverse of α(x), then you can demonstrate closure.

Properties you might find useful
(* refers to Hermitian transpose, ` to matrix inverse):
(AB)* = B*A* for all A, B
(AB)` = B`A` for all A, B
A* = A` for all unitary A

Hopefully this helps.

Answer 2

Okay, so it's unclear what you mean by U(n); it's not standard notation. It's probably not the group of unitary matrices, because the scalar matrix -1 has square 1, so alpha is then *never* injective.

I assume you mean the set of invertible elements in the mod n ring Z/nZ. Since Z/nZ is commutative, your map alpha is always a homomorphism. Since Z/nZ is a finite set, alpha is bijective if and only if it is injective. By group theory, it is injective if and only if its kernel is trivial. So you need to figure out for which n, the only square root of 1 mod n is 1. This is true, for instance, in U(2) but not in U(3).