Idempotent transformations of finite groups. (Q1931835)

Let \(\mathcal G\) be the category of groups and \(\Phi\colon\mathcal G\rightsquigarrow\mathcal G\) be an augmented functor that is an endofunctor with a natural transformation \(\varepsilon\colon\Phi\rightsquigarrow\mathrm{id}\), called augmentation. An augmented functor \((\Phi,\varepsilon)\) is called idempotent if the two homomorphisms \(\Phi(\varepsilon(G))\) and \((\Phi(G))\colon\Phi^2(G)\rightsquigarrow\Phi(G)\) are isomorphisms for any group \(G\). The aim of this paper is to understand how idempotent functors deform (finite) groups. The authors show that finiteness, nilpotency, solvability,\dots are preserved by idempotent functors. Let \(\Phi\) be an idempotent functor and let \(G\) be a finite group, then the authors prove that the set \(\mathrm{idem}(G)\) of isomorphism classes of \(\Phi(G)\) is finite. They enumerate the set \(\mathrm{idem}(G)\) and study the relation between idempotent functors and cellular cover. At the end, the authors give explicit examples to illustrate some results of the paper.