----- > [!proposition]+ Proposition. ([[homsets in ab are abelian groups]]) > Let $G,H$ in $\mathsf{Ab}$ be [[abelian group|abelian groups]]. Then homset $\text{Hom}_{\mathsf{Ab}}(G,H)$ is itself an [[abelian group|abelian]] [[group]] with [[binary operation]] $(\varphi, \psi) \mapsto \varphi + \psi$. > > This follows from a more general fact: if $G,H$ in $\mathsf{Grp}$ with (perhaps only) $H$ [[abelian group|abelian]], then $\text{Hom}_{\mathsf{Grp}}(G,H)$ is itself an [[abelian group|abelian]] [[group]]. > In the special case $G=H$, we get additional structure: [[endsets in ab are rings]]. ^proposition > [!proof]+ Proof. ([[homsets in ab are abelian groups]]) > We'll show the 'more general fact': that taking the identity to be the [[group homomorphism|trivial homomorphism]] and the [[binary operation]] on $\varphi, \psi \in \text{Hom}_{\mathsf{Grp}}(G,H)$ to be $\varphi + \psi$ makes $\text{Hom}_{\mathsf{Grp}}(G,H)$ into a [[group]] when $H$ is [[abelian group|abelian]]. We'll use additive notation for $H$ and multiplicative notation for $G$. > > Is this actually a [[binary operation]] (i.e., must $\varphi + \psi$ be a [[group homomorphism|homomorphism]])? Yes: > > $\begin{align} (\varphi + \psi)(g_{1}g_{2})= & \varphi(g_{1}) + \varphi(g_{2}) + \psi(g_{1}) + \psi(g_{2}) \\ = & (\varphi + \psi)(g_{1}) + (\varphi + \psi)(g_{2}), \end{align}$ > > where the last equality uses that $H$ is [[abelian group|abelian]]. [[associative|Associativity]] is inherited from the operation on $H$, and the group inverse of $\varphi: G \to H$ is defined to be $-\varphi$. Then we're done. ^proof ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` #reformatrevisebatch01