Set of Linear Transformations under Pointwise Addition forms Abelian Group

Theorem
Let $\struct {G, +_G}$ and $\struct {H, +_H}$ be groups.

Let $\struct {R, +_R, \times_R}$ be a ring.

Let $\struct {G, +_G, \circ}_R$ and $\struct {H, +_H, \circ}_R$ be $R$-modules.

Let $\map {\mathcal L_R} {G, H}$ be the set of all linear transformations from $G$ to $H$.

Let $\oplus_H$ be the operation on $H^G$ as defined in Addition of Linear Transformations.

Then $\struct {\map {\LL_R} {G, H}, \oplus_H}$ is an abelian group.

Proof
From Structure Induced by Group Operation is Group, $\struct {H^G, \oplus_H}$ is a group

Let $\phi, \psi \in \map {\LL_R} {G, H}$.

From Addition of Linear Transformations, $\phi \oplus_H \psi \in \map {\LL_R} {G, H}$.

From Negative Linear Transformation, $-\phi \in \map {\LL_R} {G, H}$.

Thus, from the Two-Step Subgroup Test, $\struct {\map {\LL_R} {G, H}, \oplus_H}$ is a subgroup of $\struct {H^G, \oplus_H}$.