Give an example of a cyclic group with 6 generators. Give the generators, explain how you know that these are generators and that they are the only generators.
I don't even know how to begin this problem. The textbook we're using is not the easiest to understand. All we have covered in class is order, groups and cyclic groups. Is there a way to do this problem with their definitions?
HINT: You know that the identity element can’t be a generator. The simplest possibility that could conceivably work is that it’s the only non-generator, in which case the group must have order $7$. Do you know a group of order $7$? What are its generators?
A cyclic group of order $n$ has $\phi(n)$ generators. Now knowing that $p\mid n$ implies $p-1\mid \phi(n)$, we see that $n$ must not be divisible by any prime $p>7$ and also not by $p=5$. This leaves us with $n=2^a3^b7^c$. Moreover $p^{k}\mid n$ implies $p^{k-1}\mid \phi(n)$, hence $a\le 2$, $b\le 2$, $c\le 1$. To have $3\| \phi(n)$ we need either $c=1$ or $b=2$. A bit more thought shows that $c=1$ leads to $n=7$ or $n=14$, and $b=2$ leads to $n=9$ or $n=18$.
