Is there a category theoretic approach to Sylow theorems?
More generally, is there an exposition of finite group theory in terms of category theory which would include Sylow theorems and little facts of the following sort (note use of counterexamples):
$A$ is abelian iff $\ <a,b>\longrightarrow <a,b:ab=ba> \rightthreetimes\ \ A\longrightarrow 1$
$G$ is perfect, $G=[G,G]$, iff $G \not\rightarrow A$ for A abelian
finite group $H$ is soluble iff $G \not\rightarrow H$ for G perfect
(Cauchy theorem) $G$ has order prime to $p$ iff ${\Bbb Z}/p{\Bbb Z} \not\rightarrow G$
$G$ is a $p$-group iff $H \not\rightarrow G$ for groups $H$ of order prime to $p$
(Feit-Thompson theorem) ${\Bbb Z}/2{\Bbb Z} \not\rightarrow H$ implies $G \not\rightarrow H$ for G perfect
Here $G \not\rightarrow H $ means there is no non-trivial homomorphism from $G$ into $H$.
For homomorphisms $f:A\longrightarrow B$ and $g:C\longrightarrow D$, the lifting property $f \rightthreetimes\ \ g $ means that for each homorphisms $u:A\longrightarrow C$ and $d:B\longrightarrow D$ such that $fd=ug$, there is a morphism $ d': B\longrightarrow C$ such that $u=fd'$ and $d=d'g$. For example, $G \not\rightarrow H $ iff $G\longrightarrow 1 \rightthreetimes\ H \longrightarrow 1 $ iff $1\longrightarrow H \rightthreetimes\ 1 \longrightarrow H $.
