# 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 $\eqclass 3 7^n = \eqclass 1 7$:

\(\displaystyle \eqclass 3 7^2\) | \(=\) | \(\displaystyle \eqclass {3 \times 3} 7\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \eqclass 9 7\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \eqclass 2 7\) | |||||||||||

\(\displaystyle \eqclass 3 7^3\) | \(=\) | \(\displaystyle \eqclass 3 7^2 \times \eqclass 3 7\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \eqclass {2 \times 3} 7\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \eqclass 6 7\) | |||||||||||

\(\displaystyle \eqclass 3 7^4\) | \(=\) | \(\displaystyle \eqclass 3 7^3 \times \eqclass 3 7\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \eqclass {6 \times 3} 7\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \eqclass {18} 7\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \eqclass 4 7\) | |||||||||||

\(\displaystyle \eqclass 3 7^5\) | \(=\) | \(\displaystyle \eqclass 3 7^4 \times \eqclass 3 7\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \eqclass {4 \times 3} 7\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \eqclass {12} 7\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \eqclass 5 7\) | |||||||||||

\(\displaystyle \eqclass 3 7^6\) | \(=\) | \(\displaystyle \eqclass 3 7^5 \times \eqclass 3 7\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \eqclass {5 \times 3} 7\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \eqclass {15} 7\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \eqclass 1 7\) |

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.

$\blacksquare$

## Sources

- 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): Chapter $6$: An Introduction to Groups: Exercise $16$