Order of Conjugate Element equals Order of Element/Corollary

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 $x \circ x^{-1} = e$.

$\blacksquare$