Reduced Residues Modulo 5 under Multiplication form Cyclic Group
Theorem
Let $\struct {\Z'_5, \times_5}$ denote the multiplicative group of reduced residues modulo $5$.
Then $\struct {\Z'_5, \times_5}$ is cyclic.
Proof
From Reduced Residue System under Multiplication forms Abelian Group it is noted that $\struct {\Z'_5, \times_5}$ is a group.
It remains to be shown that $\struct {\Z'_5, \times_5}$ is cyclic.
It will be demonstrated that:
- $\gen {\eqclass 2 5} = \struct {\Z'_5, \times_5}$
That is, that $\eqclass 2 5$ is a generator of $\struct {\Z'_5, \times_5}$.
We note that $\eqclass 1 5$ is the identity element of $\struct {\Z'_5, \times_5}$.
Thus successive powers of $\eqclass 2 5$ are taken, until $n \in \Z$ is found such that $\eqclass 2 5^n = \eqclass 1 5$:
| \(\ds \eqclass 2 5^2\) | \(=\) | \(\ds \eqclass {2 \times 2} 5\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \eqclass 4 5\) | ||||||||||||
| \(\ds \eqclass 2 5^3\) | \(=\) | \(\ds \eqclass 2 5^2 \times \eqclass 2 5\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \eqclass {2 \times 4} 5\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \eqclass 8 5\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \eqclass 3 5\) | ||||||||||||
| \(\ds \eqclass 2 5^4\) | \(=\) | \(\ds \eqclass 2 5^3 \times \eqclass 2 5\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \eqclass {3 \times 2} 5\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \eqclass 6 5\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \eqclass 1 5\) |
All elements of $\struct {\Z'_5, \times_5}$ are seen to be in $\gen {\eqclass 2 5}$.
Hence the result by definition of cyclic group.
$\blacksquare$