# number theory – Is \$mathbb Z _p^*={ 1, 2, 3, … , p-1 }\$ a cyclic group?

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 ?