Multiplicative Group of Reduced Residues Modulo 7 is Cyclic

Theorem
Let $\struct {\Z'_7, \times_7}$ denote the multiplicative group of reduced residues modulo $7$.

Then $\struct {\Z'_7, \times_7}$ is cyclic.

Proof
From Reduced Residue System under Multiplication forms Abelian Group‎ it is noted that $\struct {\Z'_7, \times_7}$ is a group.

It remains to be shown that $\struct {\Z'_7, \times_7}$ is cyclic.

It will be demonstrated that:
 * $\gen {\eqclass 3 7} = \struct {\Z'_7, \times_7}$

That is, that $\eqclass 3 7$ is a generator of $\struct {\Z'_7, \times_7}$.

We note that $\eqclass 1 7$ is the identity element of $\struct {\Z'_7, \times_7}$.

Thus successive powers of $\eqclass 3 7$ are taken, until $n \in \Z$ is found such that $\eqclass 3 7^n = \eqclass 1 7$:

All elements of $\struct {\Z'_7, \times_7}$ are seen to be in $\gen {\eqclass 3 7}$.

Hence the result by definition of cyclic group.