Reply To: Maths

[Harry]

Any element of Z_n has the form m.1, and if f is an automorphism then f(m)=f(m.1)=m.f(1), so it is defined by its value on 1. Since every element has to be a power of f(1), if f is bijective f(1) must generate Z_n, so in particular there is some a such that a.f(1)=1 in Zn, or, equivalently a.f(1)+b.n=1 for some a,b in Z. It follows that f(1) has to be coprime to n. Conversely if f(1) is coprime to n then f(1) generates all of Z_n and the map f(m) = m.f(1) defines an automorphism. Finally if f and g are automorphisms then so is f composed with g, and (fg)(1) = f(g(1))=g(1).f(1) mod n, so composition of automorphisms agrees with multiplication mod n. Hence, as you suspected, the automorphism group of Z_n is just the group of units, whether or not n is prime.