Order of Conjugate

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Theorem

Let $\left({G, \circ}\right)$ be a group whose identity is $e$.

Then

$\forall a, x \in \left({G, \circ}\right): \left|{x \circ a \circ x^{-1}}\right| = \left|{a}\right|$

where $\left|{a}\right|$ is the order of $a$ in $G$.

$\forall a, x \in \left({G, \circ}\right): \left|{x \circ a}\right| = \left|{a \circ x}\right|$

Then $a^k = e$, and $\forall n \in \N^*: n < k \implies a^n \ne e$.

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle \left({x \circ a \circ x^{-1} }\right)^k$	$=$	$\displaystyle x \circ a^k \circ x^{-1}$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Power of Conjugate
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle x \circ e \circ x^{-1}$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle x \circ x^{-1}$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle e$	$\displaystyle $	$\displaystyle $	$\displaystyle $

Thus $\left|{x \circ a \circ x^{-1}}\right| \le \left|{a}\right|$.

Then $x \circ a^n \circ x^{-1} = x \circ y \circ x^{-1}$.

If $x \circ y = e$, then $x \circ a^n \circ x^{-1} = x^{-1}$.

If $y \circ x^{-1} = e$, then $x \circ a^n \circ x^{-1} = x$.

So $a^n \ne e \implies x \circ a^n \circ x^{-1} = \left({x \circ a \circ x^{-1}}\right)^n \ne e$.

Thus $\left|{x \circ a \circ x^{-1}}\right| \ge \left|{a}\right|$, and the result follows.

$\blacksquare$

From the main result, putting $a \circ x$ for $a$:

$\left|{x \circ \left({a \circ x}\right) \circ x^{-1}}\right| = \left|{a \circ x}\right|$

from which the result follows by $x \circ x^{-1} = e$.

$\blacksquare$

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \left({x \circ a \circ x^{-1} }\right)^k\)	\(=\)	\(\displaystyle x \circ a^k \circ x^{-1}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Power of Conjugate
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle x \circ e \circ x^{-1}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle x \circ x^{-1}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle e\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)