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.

$\blacksquare$