Exercise 2.3.10 (Elements of $\langle \overline{30} \rangle \subset \mathbb{Z}/54\mathbb{Z}$ and their orders.)

Question

What is the order of $\overline{30}$ in $\mathbb{Z}/54\mathbb{Z}$? Write out all the elements and their orders in $\langle \overline{30} \rangle$.

Richard Ganaye · Accepted Answer

\begin{proof} 
For every integer $k$,
\begin{align*}
k \overline{30} = \overline{0} &\iff 54 \mid 30 k\
&\iff 9 \mid 5 k\
&\iff 9 \mid k,
\end{align*}
because $9 \wedge 5 = 1$.

This shows that the order of $\overline{30} \in \mathbb{Z}/54\mathbb{Z}$ is $9$:
$$|\overline{30}| = 9.$$
Therefore the subgroup of $\mathbb{Z}/54\mathbb{Z}$ generated by $\overline{30}$ is
\begin{align*}
\langle \overline{30} angle &= \{\overline{0}, \overline{30}, 2\cdot \overline{30}, 3\cdot \overline{30}, 4\cdot \overline{30}, 5\cdot \overline{30}, 6\cdot \overline{30}, 7\cdot \overline{30}, 8\cdot \overline{30}\}\qquad (9 \cdot \overline{30} = \overline{0})\
&=\{\overline{0}, \overline{30}, \overline{6},\overline{36}, \overline{12}, \overline{42}, \overline{18}, \overline{48}, \overline{24}\}.
\end{align*}
By Proposition 5, for all integers $k$,
$$ |k \cdot \overline{30}| = \frac{9}{k\wedge 9}.$$
The respective orders of $\overline{0}, \overline{30}, \overline{6},\overline{36}, \overline{12}, \overline{42}, \overline{18}, \overline{48}, \overline{24}$ are
$1, 9, 9, 3, 9, 9, 3, 9, 9$.
\end{proof}

With Sagemaths:
\begin{verbatim}
sage: Zn = Integers(54)
sage: a = Zn(30)
sage: a.additive_order()
9
sage: [k*a for k in range(9)]
[0, 30, 6, 36, 12, 42, 18, 48, 24]
sage: [(k*a).additive_order() for k in range(9)]
[1, 9, 9, 3, 9, 9, 3, 9, 9]
\end{verbatim}

Exercise 2.3.10 (Elements of $\langle \overline{30} \rangle \subset \mathbb{Z}/54\mathbb{Z}$ and their orders.)

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Exercise 2.3.10 (Elements of $\langle \overline{30} \rangle \subset \mathbb{Z}/54\mathbb{Z}$ and their orders.)

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