Order of Conjugate Element equals Order of Element/Corollary
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Corollary to Order of Conjugate Element equals Order of Element
Let $\struct {G, \circ}$ be a group whose identity is $e$.
Then:
- $\forall a, x \in \struct {G, \circ}: \order {x \circ a} = \order {a \circ x}$
where $\order a$ denotes the order of $a$ in $G$.
Proof
From Order of Conjugate Element equals Order of Element, putting $a \circ x$ for $a$:
- $\order {x \circ \paren {a \circ x} \circ x^{-1} } = \order {a \circ x}$
from which the result follows by $x \circ x^{-1} = e$.
$\blacksquare$
Sources
- 1964: Walter Ledermann: Introduction to the Theory of Finite Groups (5th ed.) ... (previous) ... (next): Chapter $\text {I}$: The Group Concept: Chapter $1$: The Group Concept: Examples: $(5)$
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Subgroups and Cosets: $\S 41 \delta$
- 1978: John S. Rose: A Course on Group Theory ... (previous) ... (next): $0$: Some Conventions and some Basic Facts: Exercise $4$
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $6$: An Introduction to Groups: Exercise $12 \ \text {(i)}$
- 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 3.4$: Cyclic groups: Exercise $6$