Example 8.3:

Between the categories ${\bf Gr}$ and ${\bf AbGr}$ we define a map as follows. For $G \in {\bf Gr}$ we denote by $A_{G}$ its abelianization namely the quotient group:

$$A_{G} = G/[G, G].$$

The quotient is an abelian group and so belongs to ${\bf AbGr}$. For example if we take $G = S_n$ the symmetric group on $n$ letters then its abelianization is the cyclic group of order two (why?). If $G$ and $H$ are two groups and $f:G\longrightarrow H$ is a group homomorphism then

$$\eta_{H}\circ f : G \longrightarrow H/[H, H]$$

is a group homomorphism into an abelian group where $\eta_H$ is the quotient map $H \longrightarrow H/[H,H]$. The kernel of $\eta_{H}\circ f$ must contain all the commutators and so defines a group homomorphism

$$\tilde{f} : G/[G, G] \longrightarrow H/[H, H]$$

$$\overline{x} \mapsto \overline{f(x)},\phantom{XXX}$$

where the bar over $x$ denotes the residue class of $x$ in the quotient. Thus to each object $G$ of ${\bf Gr}$ we have assigned a unique object of ${\bf AbGr}$ namely the abelianization $G/[G, G]$ and to each morphism $f \in$   Mor$(G, H)$ we have associated a unique morphism $\tilde f$. The following properties are quite clear:

(i) If $f \in$   Mor$(G, H)$ and $g \in$   Mor$(H, K)$ then

$$\widetilde{g\circ f} = \tilde g \circ \tilde f$$

(ii) For any group $G$,

$$\widetilde{\mbox{id}_G} = \mbox{id}_{G/[G,G]}$$

This is an example of a covariant functor from one category to another.