$\mathbb{Z}^*_m = \{a \in \mathbb{Z}_m | \gcd(a, m) = 1\}$. As $\mathbb{Z}^*_p$ is cyclic when $p$ is prime the group contains at least 1 generator. Can we say anything else about the number of generators in such a group?
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Number of generators given by Euler $\phi$ function. – Yadati Kiran Nov 27 '18 at 15:12
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But doesn't the Euler function return the cardinality of the group $\mathbb{Z}^*_p$? – Ketho Nov 27 '18 at 15:29
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Yes it does. I also gives number of integers relatively prime to $n$. https://math.stackexchange.com/questions/2155137/cyclic-group-generators-of-order-n?noredirect=1&lq=1 – Yadati Kiran Nov 27 '18 at 15:29
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Where is my mistake in this example: $\phi(19) = 18$ and the generators of $\mathbb{Z}^*_{19}$ are 2, 3, 10, 13, 14 and 15. – Ketho Nov 27 '18 at 15:35
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2The multiplicative group has 18 elements and $\phi(18)=6$ – Empy2 Nov 27 '18 at 15:59