Hey this question is going to sound really basic and dumb but I'm new to abstract algebra and not very good at this stuff. When I am talking about Z*(n), to be clear, I mean the multiplicative group of integers mod n, or the way I'm always thinking about it, the units of Z(n). (I'm not sure what the actual name for Z(n) is, but it's the equivalence classes of the integers mod n, so for example, Z(3) = {[0],[1],[2]}, Z(4) = {[0],[1],[2],[3]}, etc.)(adsbygoogle = window.adsbygoogle || []).push({});

Now, it seems like Z*(n) always has some generator. But I don't want to just assume that's the case. I understand that Z*(n) is cyclic, but does every cyclic group have a generator? (or is this just part of the definition of being cyclic - that it's a group built up by some generator) Is there a theorem that says Z*(n) is cyclic? Also, how can I tell how many generators of Z*(n) there are? For example when I'm building up Dirichlet tables for Z*(n) I notice for example Z*(5) has two generators: [2] and [3], while some of the others only have a single generator. Is there any theorem that tells me exactly how many generators there are?

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# Does Z*(n) always have a generator?

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