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Throughout this paper, all groups are finite. Let $sigma ={sigma_{i} | iin I }$ be some partition of the set of all primes $Bbb{P}$. If $n$ is an integer, the symbol $sigma (n)$ denotes the set ${sigma_{i} |sigma_{i}cap pi (n) e emptyset }$. The integers $n$ and $m$ are called $sigma$-coprime if $sigma (n)cap sigma (m)=emptyset$. Let $t > 1$ be a natural number and let $mathfrak{F}$ be a class of groups. Then we say that $mathfrak{F}$ is $Sigma_{t}^{sigma}$-closed provided $mathfrak{F}$ contains each group $G$ with subgroups $A_{1}, ldots , A_{t}in mathfrak{F}$ whose indices $|G:A_{1}|$, $ldots$, $|G:A_{t}|$ are pairwise $sigma$-coprime. In this paper, we study $Sigma_{t}^{sigma}$-closed classes of finite groups.
The first examples of formations which are arboreous (and therefore Hall) but not freely indexed (and therefore not locally extensible) are found. Likewise, the first examples of solvable formations which are freely indexed and arboreous (and therefo
Let $G$ be a finite group and $sigma$ a partition of the set of all? primes $Bbb{P}$, that is, $sigma ={sigma_i mid iin I }$, where $Bbb{P}=bigcup_{iin I} sigma_i$ and $sigma_icap sigma_j= emptyset $ for all $i e j$. If $n$ is an integer, we write $s
Let $G$ be a finite group and $sigma ={sigma_{i} | iin I}$ some partition of the set of all primes $Bbb{P}$, that is, $sigma ={sigma_{i} | iin I }$, where $Bbb{P}=bigcup_{iin I} sigma_{i}$ and $sigma_{i}cap sigma_{j}= emptyset $ for all $i e j$. We s
Let $sigma ={sigma_{i} | iin I}$ be a partition of the set $Bbb{P}$ of all primes and $G$ a finite group. A chief factor $H/K$ of $G$ is said to be $sigma$-central if the semidirect product $(H/K)rtimes (G/C_{G}(H/K))$ is a $sigma_{i}$-group for some
We construct an analogue of the normaliser decomposition for p-local finite groups (S,F,L) with respect to collections of F-centric subgroups and collections of elementary abelian subgroups of S. This enables us to describe the classifying space of a