Andrea Lucchini, Daniele Nemmi
Given a formation $\mathfrak F$, we consider the graph whose vertices are the elements of $G$ and where two vertices $g,h\in G$ are adjacent if and only if $\langle g,h \rangle \notin\mathfrak F$. We are interested in the two following questions. Is the set of the isolated vertices of this graph a subgroup of $G$? Is the subgraph obtained by deleting the isolated vertices a connected graph?
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Given a class $\mathfrak F$ of finite groups, we consider the graph $\widetildeΓ_{\mathfrak F}(G)$ whose vertices are the elements of $G$ and where two vertices $g,h\in G$ are adjacent if and only if $\langle g,h\rangle\notin\mathfrak F$. Moreover we denote by $\mathcal{I}_{\mathfrak F}(G)$ the set of the isolated vertices of $\widetildeΓ_{\mathfrak F}(G).$ We address the following question: to what extent the fact that $\mathcal{I}_{\mathfrak F}(G)$ is a subgroup of $H$ for any $H\leq G,$ implies that the graph $Γ_{\mathfrak F}G)$ obtained from $\widetildeΓ_{\mathfrak F}(G)$ by deleting the i…
For a wide family of formations $\mathfrak{F}$ it is proved that the $ \mathfrak{F}$-residual of a permutation finite group can be computed in a polynomial time. Moreover, if in the previous case $\mathfrak{F}$ is hereditary, then an $\mathfrak{F}$-subnormality of a subgroup can be checked in a polynomial time.
In this paper some properties of the $\mathfrak{F}^*$-hypercenter of a finite group are studied where $\mathfrak{F}^*$ is the class of all finite quasi-$\mathfrak{F}$-groups for a hereditary saturated formation $\mathfrak{F}$ of finite groups. In particular, it is shown that the quasinilpotent hypercenter of a finite group coincides with the intersection of all maximal quasinilpotent subgroups.
Let $\mathfrak{H}$ be a Fitting class and $\mathfrak{F}$ a formation. We call a subgroup $\mathcal{N}_{\mathfrak{H},\mathfrak{F}}(G)$ of a finite group $G$ the $\mathfrak{H}$-$\mathfrak{F}$-norm of $G$ if $\mathcal{N}_{\mathfrak{H},\mathfrak{F}}(G)$ is the intersection of the normalizers of the products of the $\mathfrak{F}$-residuals of all subgroups of $G$ and the $\mathfrak{H}$-radical of $G$. Let $π$ denote a set of primes and let $\mathfrak{G}_π$ denote the class of all finite $π$-groups. We call the subgroup $\mathcal{N}_{\mathfrak{G}_π,\mathfrak{F}}(G)$ of $G$ the $π\mathfrak{F}$-norm o…
Let $\mathfrak F$ be a class of groups. A group $G$ is called $ca$-$\mathfrak F$-group if its every non-abelian chief factor is simple and $H/K \leftthreetimes C_G(H/K) \in \mathfrak F$ for every abelian chief factor $H/K$ of $G$. In this paper, we investigate the structure of a finite $ca$-$\mathfrak F$-group. Properties of mutually permutable products of finite $ca$-$\mathfrak F$-groups are studied.
We describe the structure of finite groups with $\mathfrak{F}$-subnormal or self-normalizing primary cyclic subgroups when $\mathfrak{F}$ is a subgroup-closed saturate superradical formation containing all nilpotent groups. We prove that groups with absolutely $\mathfrak{F}$-subnormal or self-normalizing primary cyclic subgroups are soluble when $\mathfrak{F}$ is a subgroup-closed saturate formation containing all nilpotent groups.
Let $π$ be a set of primes and $\mathfrak{F}$ be a formation. In this article a properties of the class ${\rm w}^{*}_π\mathfrak{F}$ of all groups $G$, such that $π(G)\subseteq π(\mathfrak{F})$ and the normalizers of all Sylow $p$-subgroups of $G$ are $\mathfrak{F}$-subnormal in $G$ for every $p\inπ\capπ(G)$ are investigated. It is established that ${\rm w}^{*}_π\mathfrak{F}$ is a formation. Some hereditary saturated formations $\mathfrak{F}$ for which ${\rm w}^{*}_π\mathfrak{F}=\mathfrak{F}$ are founded.
A theorem of Dolfi, Herzog, Kaplan, and Lev \cite[Thm.~C]{DHKL} asserts that in a finite group with trivial Fitting subgroup, the size of the soluble residual of the group is bounded from below by a certain power of the group order, and that the inequality is sharp. Inspired by this result and some of the arguments in \cite{DHKL}, we establish the following generalisation: if $\mathfrak{X}$ is a subgroup-closed Fitting formation of full characteristic which does not contain all finite groups and $\overline{\mathfrak{X}}$ is the extension-closure of $\mathfrak{X}$, then there exists an (optimal…
On the intersection of $\mathfrak{F}$-maximal subgroups of a finite group
本文研究了有限群G的所有F-极大子群交集Int_F(G)的性质，证明了Int_F(G/Int_F(G))≈1当且仅当F包含所有其所有F-子群均为F-子正规的群，推导了Skiba和Beidleman等人的相关结论。
Let $\mathfrak{J}$ be a class of non-abelian simple groups and $\mathfrak{X}$ be a class of groups. A chief factor $H/K$ of a group $G$ is called $\mathfrak{X}$-central in $G$ provided $(H/K)\rtimes G/C_G(H/K)\in\mathfrak{X}$. We say that $G$ is a $\mathfrak{J}cs$-$\mathfrak{X}$-\emph{group} if every chief $\mathfrak{X}$-factor of $G$ is $\mathfrak{X}$-central and other chief factors of $G$ are simple $\mathfrak{J}$-groups. We use $\mathfrak{X}_{\mathfrak{J}cs}$ to denote the class of all $\mathfrak{J}cs$-$\mathfrak{X}$-groups. A subgroup $U$ of a group $G$ is called $\mathfrak{X}$-\emph{maxim…
We investigate the intersection of normalizers and $\mathfrak{F}$-subnormalizers of different types of systems of subgroups ($\mathfrak{F}$-maximal, Sylow, cyclic primary). We described all formations $\mathfrak{F}=\underset{i\in I}\times\mathfrak{F}_{π_i}$ for which the intersection of normalizers of all $\mathfrak{F}_i$-maximal subgroups of $G$ is the $\mathfrak{F}$-hypercenter of $G$ for every group $G$. Also we described all formations $\mathfrak{F}$ for which the intersection of $\mathfrak{F}$-subnormalizers of all Sylow (cyclic primary) subgroups of $G$ is the $\mathfrak{F}$-hypercenter …
Let $\mathfrak{F}$ be a class of finite groups and $G$ a finite group. Let ${\cal L}_{\mathfrak{F}}(G)$ be the set of all subgroups $A$ of $G$ with $A^{G}/A_{G}\in \mathfrak{F}$. A chief factor $H/K$ of $G$ is $\mathfrak{F}$-central in $G$ if $(H/K)\rtimes (G/C_{G}(H/K)) \in\mathfrak{F}$. We study the structure of $G$ under the hypothesis that every chief factor of $G$ between $A_{G}$ and $A^{G}$ is $\mathfrak{F}$-central in $G$ for every subgroup $A\in {\cal L}_{\mathfrak{F}}(G)$. As an application, we prove that a finite soluble group $G$ is a $PST$-group if and only if $A^{G}/A_{G}\leq Z_{\…
We present an explicit structure for the Baer invariant of a finitely generated abelian group with respect to the variety $[\mathfrak{N}_{c_1},\mathfrak{N}_{c_2}]$, for all $c_2\leq c_1\leq 2c_2$. As a consequence we determine necessary and sufficient conditions for such groups to be $[\mathfrak{N}_{c_1},\mathfrak{N}_{c_2}]$-capable. We also show that if $c_1\neq 1\neq c_2$, then a finitely generated abelian group is $[\mathfrak{N}_{c_1},\mathfrak{N}_{c_2}]$-capable if and only if it is capable. Finally we show that $\mathfrak{S}_2$-capability implies capability but there is a finitely generat…
For a formation $\mathfrak{F}$ of finite groups, tight connections are established between the pro-$\mathfrak{F}$-topology of a finitely generated free group $F$ and the geometry of the Cayley graph $Γ(\hat{F_{\mathfrak{F}}})$ of the pro-$\mathfrak{F}$-completion $\hat{F_{\mathfrak {F}}}$ of $F$. For example, the Ribes--Zalesskii-Theorem is proved for the pro-$\mathfrak{F}$-topology of $F$ in case $Γ(\hat{F_{\mathfrak F}})$ is a tree-like graph. All these results are established by purely geometric proofs, without the use of inverse monoids which were indispensable in earlier papers, thereby g…
This paper gives methods to describe the adjoint orbits of $\mathbf{G}(\mathfrak{o}_r)$ on $\mathrm{Lie}(\mathbf{G})(\mathfrak{o}_r)$ where $\mathfrak{o}_r=\mathfrak{o}/\mathfrak{p}^r$ ($r\in\mathbb{N}$) is a finite quotient of the localization $\mathfrak{o}$ of the ring of integers of a number field at a prime ideal $\mathfrak{p}$ and $\mathbf{G}$ is a closed $\mathbb{Z}$-subgroup scheme of $\mathrm{GL}_{n}$ for an $n\in\mathbb{N}$ and such that the Lie ring $\mathrm{Lie}(\mathbf{G})(\mathfrak{o})$ is quadratic.. The main result is a classification of the adjoint orbits in $\mathrm{Lie}(\math…
Formations of Finite Groups in Polynomial Time: the $\mathfrak{F}$-Radical
本文针对Baer-局部Fitting构造提出计算置换群$\mathfrak{F}$-根的多项式时间算法，并扩展至可解群的原始饱和构造及群相关长度的计算。
Let $\frak{F}$ be a class of finite groups. A subgroup $H$ of a finite group $G$ is said to be $\mathfrak{F_{\mathrm s}}$-quasinormal in $G$ if there exists a normal subgroup $T$ of $G$ such that $HT$ is $s$-permutable in $G$ and $(H\cap T)H_G/H_G$ is contained in the $\frak{F}$-hypercenter $Z_\infty^\mathfrak{F}(G/H_G)$ of $G/H_G$. In this paper, we investigate further the influence of $\mathfrak{F_{\mathrm s}}$-quasinormality of some subgroups on the structure of finite groups. New characterization of some classes of finite groups are obtained.
Let $\mathfrak{F}$ be a formation and $G$ a finite group. A subgroup $H$ of $G$ is said to be weakly $\mathfrak{F}_{s}$-quasinormal in $G$ if $G$ has an $S$-quasinormal subgroup $T$ such that $HT$ is $S$-quasinormal in $G$ and $(H\cap T)H_{G}/H_{G}\leq Z_{\mathfrak{F}}(G/H_{G})$, where $Z_{\mathfrak{F}}(G/H_{G})$ denotes the $\mathfrak{F}$-hypercenter of $G/H_{G}$. In this paper, we study the structure of finite groups by using the concept of weakly $\mathfrak{F}_{s}$-quasinormal subgroups.
Formations of Finite Groups in Polynomial Time: the $\mathfrak{F}$-Hypercenter
本文证明了一大类群系（含 Baer-局部群系）的 $\mathfrak{F}$-超中心可在多项式时间内计算，并给出了多种具体群类的算法及部分极大子群交的计算方法。
A note on meta and para-$\mathfrak{Nil}$-Hamiltonian groups_v3
本文研究亚-$\mathfrak{Nil}$-哈密顿群与仿-$\mathfrak{Nil}$-哈密顿群在包含有限指数可解子群或有限阶非幂零子群条件下，其自身或导子群的有限性。
Let $\frak {F}$ be a class of group. A subgroup $A$ of a finite group $G$ is said to be $K$-$\mathfrak{F}$-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 $A_{i}/(A_{i-1})_{A_{i}} \in \mathfrak{F}$ for all $i=1, \ldots , n$. A formation $\frak {F}$ is said to be $K$-lattice provided in every finite group $G$ the set of all its $K$-$\mathfrak{F}$-subnormal subgroups forms a sublattice of the lattice of all subgroups of $G$. In this paper we consider some new applications of the theory of $K$-lattic…
Let $\mathfrak{X}$ be a class of finite groups closed under taking subgroups, homomorphic images and extensions. It is known that if $A$ is a normal subgroup of a finite group $G$ then the image of an $\mathfrak{X}$-maximal subgroup $H$ of $G$ in $G/A$ is not, in general, $\mathfrak{X}$-maximal in $G/A$. We say that the reduction $\mathfrak{X}$-theorem holds for a finite group $A$ if, for every finite group $G$ that is an extension of $A$ (i. e. contains $A$ as a normal subgroup), the number of conjugacy classes of $\mathfrak{X}$-maximal subgroups in $G$ and $G/A$ is the same. The reduction $\…
Let for a prime $p$, $\mathfrak{X}$ (respectively $\mathfrak{Y}$) be the class of all $p$-biprimitively finite (respectively periodic $p$-conjugatively biprimitively finite) groups and $G\in \mathfrak{X}$ (respectively $G\in \mathfrak{Y}$), $V$ be a periodic subgroup of $G$ having an ascending series of normal in $G$ subgroups such that each its factor is an almost layer-finite group or a locally graded group of finite special rank, or a $WF$-group with $min-q$ on all primes $q$. We prove that $G/V \in \mathfrak{X}$ (respectively $G/V\in \mathfrak{Y}$). Also some interesting and useful prelimi…
In this paper we study the groups all whose maximal or all Sylow subgroups are $K$-$\mathfrak{F}$-subnormal in their product the with generalizations of the Fitting subgroup $\mathrm{F}^*(G)$ and $\mathrm{\tilde F}(G)$. We prove that a hereditary formation $\mathfrak{F}$ contains every group all whose Sylow subgroups are $K$-$\mathfrak{F}$-subnormal in their product with $\mathrm{F}^*(G)$ if and only if $\mathfrak{F}$ is the class of all $σ$-nilpotent groups for some partition $σ$ of the set of all primes. We obtain a new characterization of the $σ$-nilpotent hypercenter, i.e. the $\mathfrak{F…
A test for a local formation of finite groups to be a formation of soluble groups with the Shemetkov property
本文针对Shemetkov提出的寻找特定有限群局部构成的问题，基于极小非群可解的已知解，提出了一种多项式时间检查方法，用于验证有界局部构成是否为具有Shemetkov性质的可解群构成。
Let ${\mathfrak{X}}$ be a class of finite groups closed under taking subgroups, homomorphic images, and extensions. Denote by ${\mathrm{k}}_{\mathfrak{X}}(G)$ the number of conjugacy classes ${\mathfrak{X}}$-maximal subgroups of a finite group $G$. The natural problem to describe up to conjugacy ${\mathfrak{X}}$-maximal subgroups of a given finite group is complicated by the fact that it is not inductive. In particular, generally speaking, the image of an ${\mathfrak{X}}$-maximal subgroup is not ${\mathfrak{X}}$-maximal in the image of a homomorphism. Nevertheless, there are group homomorphism…
All groups under consideration are finite. Let $σ=\{σ_i \mid i\in I \}$ be some partition of the set of $\mathbb{P}$, $G$ be a group, and $\mathfrak F$ be a class of groups. Then $σ(G)=\{σ_i\mid σ_i\cap π(G)\ne \emptyset\} $ and $σ(\mathfrak F)=\cup_{G\in \mathfrak F}σ(G).$ A function $f$ of the form $f:σ\to\{\text{formations of groups}\}$ is called a formation $σ$-function. For any formation $σ$-function $f$ the class $LF_σ(f)$ is defined as follows: $$ LF_σ(f)=(G \text{ is a group } \mid G=1 \text{ or } G\ne 1\ \text{ and }\ G/O_{σ_i', σ_i}(G) \in f(σ_i) \text{ for all } σ_i \in σ(G)). $$ If…
Invariant boundary distributions for finite graphs
设$Γ$是有限连通图$\mathcal G$的基本群.设$\mathfrak M$是阿贝尔群。泛覆盖树$Δ$的边界$\partialΔ$上的{ \it分布}是定义在闭集上的$\mathfrak M$ -值测度。如果$\mathfrak M$没有$χ(\mathcal G)$ -挠，则$\partialΔ$上的$Γ$ -不变分布群同构于$H_1(\mathcal G,\mathfrak M)$ .
Suppose $4|n$, $n\geq 8$, $F=F_n=\mathbb{Q}(ζ_n+\barζ_n)$, and there is one prime $\mathfrak{p}=\mathfrak{p}_n$ above $2$ in $F_n$. We study amalgam presentations for $\operatorname{PU_{2}}(\mathbb{Z}[ζ_n, 1/2])$ and $\operatorname{PSU_{2}}(\mathbb{Z}[ζ_n, 1/2])$ with the Clifford-cyclotomic group in quantum computing as a subgroup. These amalgams arise from an action of these groups on the Bruhat-Tits tree $Δ=Δ_{\mathfrak{p}}$ for $\operatorname{SL_{2}}(F_\mathfrak{p})$ constructed via the Hamilton quaternions. We explicitly compute the finite quotient graphs and the resulting amalgams for $8…
The solvable Graph of a finite-dimensional Lie Algebra
本文引入有限维李代数的可解图Γ_𝔖(L)，研究其基本性质、结构约束及实例，确定特定李代数可解图的度序列，提供计算框架，揭示与幂零图的异同，为李理论可解性研究提供新图形方法。
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