Order of Power of Group Element/Examples/Powers of Element of Order 20

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Example of Order of Power of Group Element

Let $G$ be a group.

Let $x \in G$ be such that:

$\order x = 20$

where $\order x$ denotes the order of $x$ in $G$.

Then:

\(\text {(1)}: \quad\) \(\ds \order {x^4}\) \(=\) \(\ds 5\)
\(\text {(2)}: \quad\) \(\ds \order {x^{10} }\) \(=\) \(\ds 2\)
\(\text {(3)}: \quad\) \(\ds \order {x^{12} }\) \(=\) \(\ds 5\)


Proof

From Order of Power of Group Element:

$\order {x^m} = \dfrac {\order x} {\gcd \set {m, \order x} }$


Here we have that $\order x = 20$.

Thus:

\(\text {(1)}: \quad\) \(\ds \order {x^4}\) \(=\) \(\ds \dfrac {20} {\gcd \set {4, 20} }\)
\(\ds \) \(=\) \(\ds \dfrac {20} 4\) as $\gcd \set {4, 20} = 4$
\(\ds \) \(=\) \(\ds 5\)
\(\text {(2)}: \quad\) \(\ds \order {x^{10} }\) \(=\) \(\ds \dfrac {20} {\gcd \set {10, 20} }\)
\(\ds \) \(=\) \(\ds \dfrac {20} {10}\) as $\gcd \set {10, 20} = 10$
\(\ds \) \(=\) \(\ds 2\)
\(\text {(3)}: \quad\) \(\ds \order {x^{12} }\) \(=\) \(\ds 5\)
\(\ds \) \(=\) \(\ds \dfrac {20} 4\) as $\gcd \set {12, 20} = 4$
\(\ds \) \(=\) \(\ds 5\)

$\blacksquare$


Sources