# Todd Trimble monomorphisms in the category of groups

This page is to record a constructive proof of the following result.

###### Proposition

Every monomorphism in the category of groups is an equalizer.

###### Proof

Let $i: H \to G$ be monic, and let $\pi: G \to G/H$ be the canonical surjective function $g \mapsto g H$ . Let $A$ be the free abelian group on $G/H$ with $j: G/H \to A$ the canonical injection, and let $A^G$ denote the set of functions $f: G \to A$, with the pointwise abelian group structure inherited from $A$. This carries a $G$-module structure defined by

$(g \cdot f)(g') = f(g' g).$

For any $f \in A^G$, the function $d_f: G \to A^G$ defined by $d_f(g) = g f - f$ defines a derivation. Passing to the wreath product $A^G \rtimes G$, we have two homomorphisms $\phi, \psi: G \rightrightarrows A^G \rtimes G$ defined by $\phi(g) := (d_{j \pi}(g), g)$ and $\psi(g) := (0, g)$. I claim that $i: H \to G$ is the equalizer of the pair $\phi, \psi$. For,

$\array{ d_{j\pi}(g) = 0 & \text{iff} & (\forall_{g': G})\; g\cdot j\pi(g') = j\pi(g') \\ & \text{iff} & (\forall_{g': G})\; j\pi(g' g) = j\pi(g') \\ & \text{iff} & (\forall_{g': G})\; j(g' g H) = j(g' H) \\ & \text{iff} & (\forall_{g': G})\; g' g H = g' H \\ & \text{iff} & g H = H \\ & \text{iff} & g \in H. }$

(All we needed was some injection $j: G/H \to A$ into an abelian group; I chose the canonical one.)

###### Remark

This proof can be adapted to show that monomorphisms in the category of finite groups (group objects in $FinSet$) are also equalizers. All that needs to be modified is the choice of $A$, which we could take to be the $\mathbb{F}_2$-vector space freely generated by $G/H$.

###### Corollary

The category of groups is balanced: every epic mono is an isomorphism.

###### Proof

This follows because an epic equalizer is an equalizer of two maps that are the same, hence an isomorphism.

###### Corollary

Every epimorphism in the category of groups is a coequalizer.

###### Proof

Since every morphism $f: G \to H$ factors as a regular epi $p: G \to G/\ker(f)$ followed by a mono $i$, having $f$ epic implies $i$ is a epic mono. Epic monos $i$ being isomorphisms, $f$ is then forced to be regular epic as well.

Revised on January 1, 2020 at 12:33:27 by Todd Trimble