Multiplicative Group of Reduced Residues Modulo 7 is Cyclic

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

\(\ds \eqclass 3 7^2\) \(=\) \(\ds \eqclass {3 \times 3} 7\)
\(\ds \) \(=\) \(\ds \eqclass 9 7\)
\(\ds \) \(=\) \(\ds \eqclass 2 7\)
\(\ds \eqclass 3 7^3\) \(=\) \(\ds \eqclass 3 7^2 \times \eqclass 3 7\)
\(\ds \) \(=\) \(\ds \eqclass {2 \times 3} 7\)
\(\ds \) \(=\) \(\ds \eqclass 6 7\)
\(\ds \eqclass 3 7^4\) \(=\) \(\ds \eqclass 3 7^3 \times \eqclass 3 7\)
\(\ds \) \(=\) \(\ds \eqclass {6 \times 3} 7\)
\(\ds \) \(=\) \(\ds \eqclass {18} 7\)
\(\ds \) \(=\) \(\ds \eqclass 4 7\)
\(\ds \eqclass 3 7^5\) \(=\) \(\ds \eqclass 3 7^4 \times \eqclass 3 7\)
\(\ds \) \(=\) \(\ds \eqclass {4 \times 3} 7\)
\(\ds \) \(=\) \(\ds \eqclass {12} 7\)
\(\ds \) \(=\) \(\ds \eqclass 5 7\)
\(\ds \eqclass 3 7^6\) \(=\) \(\ds \eqclass 3 7^5 \times \eqclass 3 7\)
\(\ds \) \(=\) \(\ds \eqclass {5 \times 3} 7\)
\(\ds \) \(=\) \(\ds \eqclass {15} 7\)
\(\ds \) \(=\) \(\ds \eqclass 1 7\)


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

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