Discrete math Group - Isomorphism and Automorphism

Question

Let G be a Cyclic group

Prove or disprove: A.let $ a,b \in G \quad $ so the function $ f:G \to G,f(a^k) = b^k$ is Automorphism of G(which means G is Isomorphism to herself)

B.let a,b generators of G so the function $ f:G \to G,f(a^k) = b^k$ is Automorphism of G(which means G is Isomorphism to herself)

it think A is wrong and B is right but i don't know t prove or disprove it formally thanks

Answer 1

To disprove the first statement, find a counterexample.

Here's one: take $G = \{0,1,2\}$ with addition modulo $3$. Define $$ f(k) = f(k\cdot 0) = f(1 + \cdots + 1) = 0 + \cdots + 0 = 0 $$ That is, $f(g) = 0$ for all $g \in G$. Show that $f$ is a homomorphism but not an isomorphism.

To prove the second statement: follow the usual steps. That is, show that $f$ is a homomorphism that is both one to one (injective) and onto (surjective).