Order of Group Element equals Order of Inverse
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Theorem
Let $G$ be a group whose identity is $e$.
Then:
- $\forall x \in G: \order x = \order {x^{-1} }$
where $\order x$ denotes the order of $x$.
Proof
By Powers of Group Elements: Negative Index:
- $\paren {x^k}^{-1} = x^{-k} = \paren {x^{-1} }^k$
Hence:
\(\ds x^k\) | \(=\) | \(\ds e\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \paren {x^{-1} }^k\) | \(=\) | \(\ds e^{-1}\) | |||||||||||
\(\ds \) | \(=\) | \(\ds e\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \order {x^{-1} }\) | \(\le\) | \(\ds \order x\) | Definition of Order of Group Element |
Similarly:
\(\ds \paren {x^{-1} }^k\) | \(=\) | \(\ds e\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \paren {\paren {x^{-1} }^{-1} }^k\) | \(=\) | \(\ds e^{-1}\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds x^k\) | \(=\) | \(\ds e^{-1}\) | |||||||||||
\(\ds \) | \(=\) | \(\ds e\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \order x\) | \(\le\) | \(\ds \order {x^{-1} }\) | Definition of Order of Group Element |
A similar argument shows that if $x$ is of infinite order, then so must $x^{-1}$ be.
Hence the result.
$\blacksquare$
Sources
- 1964: Walter Ledermann: Introduction to the Theory of Finite Groups (5th ed.) ... (previous) ... (next): Chapter $\text {I}$: The Group Concept: $\S 8$: The Order (Period) of an Element: $\text{(ii)}$
- 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 1.9$: Exercise $5.6$
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $6$: An Introduction to Groups: Exercise $10$