Idempotent transformations of finite groups

Torsion homology and cellular approximation

“…It is also remarkable that this is proved using only group-theoretic tools, with no need of homotopical background. Some of the authors of the previous paper took a big step forward in [3], where it is proved (using the finiteness of the Schur multiplier for finite groups) that the number of cellular covers of finite group is always finite. If the group is moreover simple, then the non-trivial cellular covers of it are in bijective correspondence with the subgroups of the Schur multiplier that are invariant under the action of the automorphisms of the group.…”

Cellular Covers of Local Groups

“…It is helpful to think about localization and cellularization as functors that act on the whole category of groups, transforming some groups drastically, possibly killing many of them, and allowing us to focus on some special features such as torsion or divisibility. In [4] the authors studied the effect of iterating different cellularization functors on a given (finite) group, that is, looking at the "orbit" of a group under the action of all possible cellularization functors. A similar approach in the coaugmented case is probably quite difficult, as shown by the work of Rodríguez and Scevenels, [19].…”

“…Often LcellG is the trivial group. This happened in Example 2.1, and also for example when G is a finite simple group, L = L ab is abelianization (localization with respect to the homomorphism Z * Z → Z ⊕ Z) and cellG is any cellular cover of G. This comes from the fact that cellG is a perfect group,[4, Section 11].…”

On localizations of quasi-simple groups with given countable center

“…In this paper we apply Obraztsov [15] to find localizations of (quasi)-simple groups with given countable center, see Theorem 2.3 These ideas come from [11], where the existence of 2 ℵ 0 varieties of groups not closed under cellular covers is shown; this complements the fact that there are 2 ℵ 0 varieties closed under taking cellular covers [6]. Recall that cellular covers of simple groups were described in [1], see also [3].…”