In undergraduate course, the two groups which are most frequently used may be $${ 0, 1, 2, … , p-1}$$ and $${ 1, 2, … , p-1}$$ where $p$ is a prime.

The **first one** is a group under addition and in addition it is a cyclic group whose generator is $p-1$. Also we can describe it by the solution set of $x^p=1$ in ${bf C}$.

The **latter** is a group under multiplication. Fermat’s theorem implies that $$a^{p-1} =1~~~ (text{mod};;p)$$ for $ain { 1, … , p-1}$. But this is not sufficient for $

{ 1, … , p-1}$ to be cyclic.

My question is :

$$ { 1, … , p-1}$$ is cyclic ?

Thank you in advance.