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
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $6$: An Introduction to Groups: Exercise $13$