Conjugate of a Set by Identity

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Theorem

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

Let $S \subseteq G$.

$S^e = S$

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle S^e$	$=$	$\displaystyle \left\{ {y \in G: \exists x \in S: y = e \circ x \circ e^{-1} }\right\}$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Definition of Conjugate of a Set
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \left\{ {y \in G: \exists x \in S: y = x}\right\}$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Definition of Identity
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle S$	$\displaystyle $	$\displaystyle $	$\displaystyle $

$\blacksquare$

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle S^e\)	\(=\)	\(\displaystyle \left\{ {y \in G: \exists x \in S: y = e \circ x \circ e^{-1} }\right\}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of Conjugate of a Set
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \left\{ {y \in G: \exists x \in S: y = x}\right\}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Definition of Identity
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle S\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)