$f,g:\mathbb Z_5 \to S_5$ be non-trivial group homomorphisms , then $\exists \sigma \in S_5$ such that $f([1])=\sigma g([1])\sigma ^{-1}$?

Question

Let $f,g:\mathbb Z_5 \to S_5$ be non-trivial group homomorphisms , then is it true that $\exists \sigma \in S_5$ such that

$f([1])=\sigma g([1])\sigma ^{-1}$ ? Since both $f,g$ are non-trivial , I know that $|kerf|=|kerg|=1$ , so both $f,g$ are injective and their image in $S_5$ are cyclic of order $5$ and hence their images are isomorphic ; but I can't proceed further , please help . Thanks in advance

Answer 1

hint: both $f(1)$ and $g(1)$ are cycles of length $5$. When do two permutations conjugate?