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Given two prime numbers ${p, q > 2}$, where ${p=2q+1}$, I have to show that the cyclic group ${G = \mathbb{Z}_p^*}$ has ${p-1}$ generators.

I know that ${|G| = p-1 = 2q}$ and that ${a \in G}$ is a generator iff ${a^2 \neq 1~\text{mod}~p}$ and ${a^q \neq 1~\text{mod}~p}$. So there have to be ${2q}$ solutions for these two equations, but I have no idea how to show that.

Can you give me a hint?

Thomas Andrews
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Zoidberg
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    Do you mean $q-1$ generators? – anon Jun 24 '13 at 14:30
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    @hardmath That is not a duplicate. Exercise: why? – anon Jun 24 '13 at 14:36
  • @anon: Insofaras the Question here is a misstatement, then of course it is not an exact duplicate. However the ground has been fully covered before how to correctly count the number of generators for the cyclic group $Z_p^*$, $p$ prime. Perhaps there is better candidate for this role? – hardmath Jun 24 '13 at 14:50
  • Nowhere in the linked thread is the number of primitive roots calculated (which is the subject of this question) - the number is mentioned in passing in Erick's answer because counting them (here when $p=2q+1$ specifically) is rather trivial compared to efficiently finding one computationally (the subject of the linked question). I do not think there is a charitable interpretation in which the link given is a reasonable candidate for duplicate. I agree that the ground of this question has no doubt been covered elsewhere many times on MSE though, can't find any offhand. – anon Jun 24 '13 at 17:07
  • @anon: The link's comments and answers point out that the number of primitive roots (number of generators) is $\phi(\phi(p))=\phi(p-1)$. I grabbed that link (from 2012) because a more recent question asking about cyclic generators of $\mathbb{Z}_p^*$ was closed in deference to that one, and I wanted to maintain a consistent chain of duplicates. – hardmath Jun 24 '13 at 17:35
  • From 2011 here's another duplicate candidate Order of cyclic groups which asks for an explanation of why $(\mathbb{Z}/n\mathbb{Z})^{\times}$ is cyclic if and only if $n=1,2,4$ or $p^k$ or $2p^k$ for positive integer power of a prime $p$. Arturo's accepted Answer states "They first prove every prime $p$ has $\phi(p−1)$ primitive roots modulo $p$...". – hardmath Jun 24 '13 at 17:53
  • When proposing that B is a duplicate of A, I think we should make sure the ground necessary to obtain the desired conclusion to A is actually covered fully somewhere in B (comments, answers, or even the question itself) at the level of the OP to A; I do not think mentions-in-passing or reference-dropping are sufficient for this purpose unless either A is graduate/research-level, all questions in A are exactly duplicated in B, or said references are easily available (i.e. free and online in a widely used format). – anon Jun 25 '13 at 01:12
  • Systematically closing questions because they are closely related seems to be a letter-vs.-spirit issue; I like to keep in mind that the motivation behind the closure feature is to efficiently organize and in some exceptional cases preclude duplication of effort. (Most of the time I do think the only users that might care about duplication of effort are the authors of older answers.) Preventing OPs (and other readers) from receiving good guidance on perfectly satisfactory questions simply for asking too late, an inevitable de-facto consequence of unregulated or zealous closures, seems unfair. – anon Jun 25 '13 at 01:19
  • Another previous Question which explicitly answers this one: What are the generators for $\mathbb{Z}_p^*$ with p a safe prime?. Quoting from the Question: "[L]ets consider $\mathbb{Z}_p^$ with $p=2⋅q+1$ a safe prime ($p$ and $q$ have to be prime). Then $\varphi\left(p\right) = 2 \cdot q$ is the order of $\mathbb{Z}_p^$, and $\varphi\left(\varphi\left(p\right)\right) = q-1$ the number of generators in $\mathbb{Z}_p^*$." – hardmath Jun 25 '13 at 15:12

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How many generators does a cyclic group of order $n$ have? What is the order of $({\bf Z}/p{\bf Z})^\times$?

anon
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  • I feel that linking to previous Questions, where OPs and other readers can find explicit answers and references, does provide the "good guidance" you speak of above. Your one liner needs elaboration to meet that standard. – hardmath Jun 25 '13 at 15:21
  • Obviously I agree that linking to previous questions where there are explicit answers provides good guidance. Whether or not my one-liner (which is about equivalent to the background exposition in the latest linked question you gave, of which I agree is a reasonable candidate for duplicate) needs elaboration is up to the OP; it is what the edit feature and more prominently the comment-under-answers feature is for, which I occasionally deliberately exploit to let OPs try their own hand in stages. – anon Jun 25 '13 at 19:59