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Register a new userOne application of the $sigma$-local formations of finite groups
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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.
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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 therefore Hall) but not locally extensible are constructed. Some open questions are also mentioned.
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 $sigma(n)={sigma_i mid sigma_{i}cap pi (n) e emptyset }$ and $sigma (G)=sigma (|G|)$. We call a graph $Gamma$ with the set of all vertices $V(Gamma)=sigma (G)$ ($G e 1$) a $sigma$-arithmetic graph of $G$, and we associate with $G e 1$ the following three directed $sigma$-arithmetic graphs: (1) the $sigma$-Hawkes graph $Gamma_{Hsigma }(G)$ of $G$ is a $sigma$-arithmetic graph of $G$ in which $(sigma_i, sigma_j)in E(Gamma_{Hsigma }(G))$ if $sigma_jin sigma (G/F_{{sigma_i}}(G))$; (2) the $sigma$-Hall graph $Gamma_{sigma Hal}(G)$ of $G$ in which $(sigma_i, sigma_j)in E(Gamma_{sigma Hal}(G))$ if for some Hall $sigma_i$-subgroup $H$ of $G$ we have $sigma_jin sigma (N_{G}(H)/HC_{G}(H))$; (3) the $sigma$-Vasilev-Murashko graph $Gamma_{{mathfrak{N}_sigma }}(G)$ of $G$ in which $(sigma_i, sigma_j)in E(Gamma_{{mathfrak{N}_sigma}}(G))$ if for some ${mathfrak{N}_{sigma }}$-critical subgroup $H$ of $G$ we have $sigma_i in sigma (H)$ and $sigma_jin sigma (H/F_{{sigma_i}}(H))$. In this paper, we study the structure of $G$ depending on the properties of these three graphs of $G$.
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 say that $G$ is $sigma$-primary if $G$ is a $sigma _{i}$-group for some $i$. A subgroup $A$ of $G$ is said to be: ${sigma}$-subnormal in $G$ if there is a subgroup chain $A=A_{0} leq A_{1} leq cdots leq A_{n}=G$ such that either $A_{i-1}trianglelefteq A_{i}$ or $A_{i}/(A_{i-1})_{A_{i}}$ is $sigma$-primary for all $i=1, ldots, n$, modular in $G$ if the following conditions hold: (i) $langle X, A cap Z rangle=langle X, A rangle cap Z$ for all $X leq G, Z leq G$ such that $X leq Z$, and (ii) $langle A, Y cap Z rangle=langle A, Y rangle cap Z$ for all $Y leq G, Z leq G$ such that $A leq Z$. In this paper, a subgroup $A$ of $G$ is called $sigma$-quasinormal in $G$ if $L$ is modular and ${sigma}$-subnormal in $G$. We study $sigma$-quasinormal subgroups of $G$. In particular, we prove that if a subgroup $H$ of $G$ is $sigma$-quasinormal in $G$, then for every chief factor $H/K$ of $G$ between $H^{G}$ and $H_{G}$ the semidirect product $(H/K)rtimes (G/C_{G}(H/K))$ is $sigma$-primary.
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 $i=i(H/K)$. $G$ is called $sigma$-nilpotent if every chief factor of $G$ is $sigma$-central. We say that $G$ is semi-${sigma}$-nilpotent (respectively weakly semi-${sigma}$-nilpotent) if the normalizer $N_{G}(A)$ of every non-normal (respectively every non-subnormal) $sigma$-nilpotent subgroup $A$ of $G$ is $sigma$-nilpotent. In this paper we determine the structure of finite semi-${sigma}$-nilpotent and weakly semi-${sigma}$-nilpotent groups.
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 p-local finite group, before p-completion, as the homotopy colimit of a diagram of classifying spaces of finite groups whose shape is a poset and all maps are induced by group monomorphisms.
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