Inverse Mapping in Induced Structure of Homomorphism to Abelian Group

Theorem

Let $\struct {S, \circ}$ be an algebraic structure.

Let $\struct {T, \oplus}$ be an abelian group.

Let $f$ be a homomorphism from $S$ into $T$.

Let $f^*$ be the pointwise inverse of $f$.

Then $f^*$ is a homomorphism from $\struct {S, \circ}$ into $\struct {T, \oplus}$.

Proof

Let $\struct {T, \oplus}$ be an abelian group.

Let $x, y \in S$.

Then:

\(\ds \map {f^*} {x \circ y}\)

\(=\)

\(\ds \paren {\map f {x \circ y} }^{-1}\)

Definition of Pointwise Inverse

\(\ds \)

\(=\)

\(\ds \paren {\map f x \oplus \map f y}^{-1}\)

$f$ is a homomorphism

\(\ds \)

\(=\)

\(\ds \paren {\map f y \oplus \map f x}^{-1}\)

Commutativity of $\oplus$

\(\ds \)

\(=\)

\(\ds \paren {\map f x}^{-1} \oplus \paren {\map f y}^{-1}\)

Inverse of Group Product

\(\ds \)

\(=\)

\(\ds \map {f^*} x \oplus \map {f^*} y\)

Definition of Pointwise Inverse

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {II}$: New Structures from Old: $\S 13$: Compositions Induced on Cartesian Products and Function Spaces: Theorem $13.7: \ 2^\circ$