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Say I am presented with a group, e.g the integers ${1,2,3,4}$ under multiplication modulo 5. How would I go about identifying all of the characters? (I happen to know, due to availability of answers, that the characters in this case are: $$f(1) = f(2) = f(3) = f(4) = 1$$ $$f(1) = f(4) = 1; f(2) = f(3) = -1$$ $$f(1) = 1; f(2) = i; f(3) = -i; f(4) = -1$$ $$f(1) = 1; f(2) = -i; f(3) = i; f(4) = -1$$

My question is, how were those generated? And more generically, given some finite, abelian group G', is there a method to at least somewhat systematically generate its characters? Or is the approach simple brute force/inspection?

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    There's a large collection of techniques for the character theory of finite groups; [here](http://www.math.uconn.edu/~kconrad/blurbs/grouptheory/charthy.pdf) are some notes by Keith Conrad on characters of Abelian groups. – 2017-02-13
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    For $\mathbb{Z}_n$ the characters are of the form $\chi(m) = e^{2i \pi m k /n}$. This generalize easily to any finite abelian group – 2017-02-13

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