Reduced Residues Modulo 5 under Multiplication form Cyclic Group

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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 $\eqclass 2 5^n = \eqclass 1 5$:

\(\displaystyle \eqclass 2 5^2\) \(=\) \(\displaystyle \eqclass {2 \times 2} 5\)
\(\displaystyle \) \(=\) \(\displaystyle \eqclass 4 5\)
\(\displaystyle \eqclass 2 5^3\) \(=\) \(\displaystyle \eqclass 2 5^2 \times \eqclass 2 5\)
\(\displaystyle \) \(=\) \(\displaystyle \eqclass {2 \times 4} 5\)
\(\displaystyle \) \(=\) \(\displaystyle \eqclass 8 5\)
\(\displaystyle \) \(=\) \(\displaystyle \eqclass 3 5\)
\(\displaystyle \eqclass 2 5^4\) \(=\) \(\displaystyle \eqclass 2 5^3 \times \eqclass 2 5\)
\(\displaystyle \) \(=\) \(\displaystyle \eqclass {3 \times 2} 5\)
\(\displaystyle \) \(=\) \(\displaystyle \eqclass 6 5\)
\(\displaystyle \) \(=\) \(\displaystyle \eqclass 1 5\)


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.

$\blacksquare$