I have proven cyclic group to be a group, but now must prove cyclic group order n to be isomorphic to Zn. I realize I must show homomorphism, injection and surjection. But not struggling with initial equivalences. Please help.
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Suppose the operation of your group is given by $\displaystyle +$, if an $\displaystyle f$ as I described existed, what would be the image of $\displaystyle ka=a+a+a\hdots a$ ($\displaystyle k$ summands) under $\displaystyle f$, and $\displaystyle a$ is a generator so...
MAdone, this is really the whole of the idea.
if $\displaystyle f:G \to \mathbb{Z}_n$ is to be a homomorphism, we must have f(a*a) = f(a) + f(a) = 1+1. so there is really only "one" essential way to define f:
$\displaystyle f(a^k) = k$. that f is a homomorphism follows immediately from the laws of exponents.
since G is finite it suffices to show that f is surjective (clearly |G| = $\displaystyle |\mathbb{Z}_n|$ = n), and thus f is an isomorphism.