Pointwise Inverse in Induced Structure

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Theorem

Let $\struct {G, \oplus}$ be a group whose identity is $e_G$.

Let $S$ be a set.

Let $\struct {G^S, \oplus}$ be the structure on $G^S$ induced by $\oplus$.


Let $f \in G^S$.

Let $f^* \in G^S$ be the pointwise inverse of $f$:

$\forall x \in S: \map {f^*} x = \paren {\map f x}^{-1}$


Then $f^*$ is the inverse of $f$ for the pointwise operation induced on $G^S$ by $\oplus$.


Proof

Let $f \in G^S$.

\(\ds \map {\paren {f \oplus f^*} } x\) \(=\) \(\ds \map f x \oplus \map {f^*} x\)
\(\ds \) \(=\) \(\ds \map f x \oplus \paren {\map f x}^{-1}\)
\(\ds \) \(=\) \(\ds e_G\)

Similarly for $\map {\paren {f^* \oplus f} } x$.

$\blacksquare$


Sources

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