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: