There is an automorphism of $\mathbb Z_6$ which is not an inner automorphism I'm trying to show that there is an automorphism of $\mathbb Z_6$ which is not an inner automorphism. Since the generators of $\mathbb Z_6$ are 1 and 5, then we have two choices, we exclude the identity, it left only one possibility which it shouldn't be an inner automorphism, but my problem is I don't know how to prove that this automorphism isn't an inner automorphism. I need help Thanks for any help

Since this group is abelian (i.e. commutative), the inner automorphisms are all trivial. So you only need to find a nontrivial automorphism. Again, since this group is abelian, the inversion map is an automorphism (which is nontrivial in this case).