Finding all automorphisms of Z_10

[1800bigk]

Hi, I am going over some things for an exam and I have a question about the automorphisms of Z10 ={0,1,2,3...9} addition mod 10. I know the criteria for an automorphism and I know that an automorphism sends a generator to a generator. So Z10 has generators 1,3,7,9 so the automorphisms of Z10 are defined by a(1)=1, a(1)=3, a(1)=7. a(1)=9. I know those are the only automorphisms because of the properties of an isomorphism. I also know that once we know where the generators get sent we can figure out where everything else goes. My question is what if I wanted to define an explicit function like f: mapping Z10 to Z10 f(x) = ? where f is 1-1, onto and operation preserving. What could satisfy this? My book says its usually hard to figure out the function and they don't excpect us to define it but I want to know one so I can sleep better.
Would f(x)=xmod10 work?

 

[AKG]

There are 4 generators, g₁ = 1, g₂ = 3, g₃ = 7, g₄ = 9. You can define, for each i in {1,2,3,4} the function $f_i : \mathbb{Z}_{10} \to \mathbb{Z}_{10}$ by:

$$f_i(x) = x\cdot _{10}g_i$$

where $\cdot _{10}$ denotes multiplication modulo 10. I can't see why you'd want to do this though.