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First of all Im still new in Group theory, and I'm having trouble understading homomorphisms. I'm trying to solve an example and exposing my logic. Please correct me, and thanks in advance.

Indicate all homomorphisms $f:\mathbb{Z}_{27}\to S_3$, and indicate $f(1_{27}$) and $f(x_{27}$) for each one of them.

My argument is :

If $f:\mathbb{Z}_{27}\to S_3$ is a homomorphism, $f$ is determinated by $f(1)$, which can be any element of $S_3$ whose order divides $27$. Since it has to divide $27$ and we are in $S_3$ we want the elements with order $1$ and $3$. So, which elements in $S_3$ have order $1$ and $3$ ? Identity for $1$ but what about $3$? I stumble at this point.

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You should learn how elements of groups $S_n$ look like and how to determine their orders. In the case of $S_3$ you can do it yourself by hand from the definition since the group is really small.

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    When I say $S_3$ , I'm trying to be generic. This example could be by hand, but if I have to determinate for $S_7$ for example, my problem remains. – 2017-01-23
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    Then the first part of my answer is valid – learn that convenient way to represent permutations is their decomposition into independent cycles, and that order of a permutation is the least common multiple of the lengths of the cycles. – 2017-01-23