A note on localizations of perfect groups

Note on the universality and the functoriality of the perfect F-locality

In "Frobenius Categories versus Brauer Blocks" we have proved some universality of the so-called localizing functor associated with a Frobenius $P$-category $F$, where $P$ is a finite $p$-group, with respect to the coherent $F$-localities $(τ,L,π)$ such that the contravariant functor $Ker (π)$ maps any subgroup of $P$ to an Abelian $p$-group. The purpose of this Note is both to move from the localizing functor to the perfect locality associated with $F$ and to remove the Abelian hypothesis in the target. As a consequence, we get the functoriality for the perfect localities in the strongest for…

Preservation of perfectness and acyclicity; Berrick and Casacuberta's universal acyclic space localized at a set of primes

In this paper we answer negatively a question posed by Casacuberta, Farjoun, and Libman about the preservation of perfect groups under localization functors. Indeed, we show that a certain $P$-localization of Berrick and Casacuberta's universal acyclic group is not perfect. We also investigate under which conditions perfectness is preserved: For instance, we show that if the localization of a perfect group is finite then it is perfect.

关于Dyer群的导出长度

On the derived length of Dyer groups

本文通过分析Dyer图的性质，描述了拟完美Dyer群的导出长度特征。

关于具有TI Sylow $p$-子群的有限群块的一个注记

A note on blocks of finite groups with TI Sylow $p$-subgroups

本文针对特征零代数闭域上的有限群块，构造了一个函子等价但不完全等距的具体例子，从而说明函子等价并不蕴含完全等距。

Group localization and two problems of Levine

A. K. Bousfield's $H\mathbb Z$-localization of groups inverts homologically two-connected homomorphisms of groups. J. P. Levine's algebraic closure of groups inverts homomorphisms between finitely generated and finitely presented groups which are homologically two-connected and for which the image normally generates. We resolve an old problem concerning Bousfield $H\mathbb Z$-localization of groups, and answer two questions of Levine regarding algebraic closure of groups. In particular, we show that the kernel of the natural homomorphism from a group $G$ to it's Bousfield $H\mathbb Z$-localiza…