Leonid A. Kurdachenko, Igor Ya. Subbotin, Vasiliy A. Chepurdya
Let R be a ring and G a group. An R-module A is said to be artinian-by-(finite rank) if TorR(A) is artinian and A/TorR(A) has finite R-rank. The authors study ZG-modules A such that A/CA(H) is artinian-by-(finite rank) (as a Z-module) for every proper subgroup H.
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I prove the group theory analogues of some Lie and Leibniz algebra results on F-hypercentral and F-hypereccentric modules.
In this paper, we prove the existence and conjugacy of injectors of a generalized $π$-soluble groups for the Hartley class defined by a invariable Hartley function, and give a description of the structure of the injectors.
We introduce the concept of a strongly t-logarithmic t-generating set for a Z[t,t^{-1}]-module, which enables us to prove that a large class of soluble groups are not almost convex. We also prove some results about dead-end depth.
The Andrews-Curtis conjecture claims that every normally generating $n$-tuple of a free group $F_n$ of rank $n \ge 2$ can be reduced to a basis by means of Nielsen transformations and arbitrary conjugations. Replacing $F_n$ by an arbitrary finitely generated group yields natural generalizations whose study may help disprove the original and unsettled conjecture. We prove that every finitely generated soluble group satisfies the generalized Andrews-Curtis conjecture in the sense of Borovik, Lubotzky and Myasnikov. In contrast, we show that some soluble Baumslag-Solitar groups do not satisfy the…
Let $σ=\{σ_{i} | i\in I\}$ be a partition of the set of all primes $\Bbb{P}$ and $G$ a finite group. Let $σ(G)=\{σ_{i} : σ_{i}\cap π(G)\ne \emptyset$. A set ${\cal H}$ of subgroups of $G$ is said to be a complete Hall $σ$-set of $G$ if every member $\ne 1$ of ${\cal H}$ is a Hall $σ_{i}$-subgroup of $G$ for some $i\in I$ and $\cal H$ contains exactly one Hall $σ_{i}$-subgroup of $G$ for every $i$ such that $σ_{i}\in σ(G)$. We say that $G$ is $σ$-full if $G$ possesses a complete Hall $σ$-set. A complete Hall $σ$-set $\cal H$ of $G$ is said to be a $σ$-basis of $G$ if every two subgroups $A, B \…
We develop new invariants similar to the Bieri-Strebel-Neumann-Renz invariants but in the category of Bredon modules (with respect to the class of the finite subgroups of G). We prove that for virtually soluble groups of type FP_{\infty} and finite extension of the Thompson group F the new invariants coincide with the classical ones.
A remarkable result of Thompson states that a finite group is soluble if and only if its two-generated subgroups are soluble. This result has been generalized in numerous ways, and it is in the core of a wide area of research in the theory of groups, aiming for global properties of groups from local properties of two-generated (or more generally, $n$-generated) subgroups. We contribute an extension of Thompson's theorem from the perspective of factorized groups. More precisely, we study finite groups $G = AB$ with subgroups $A,\ B$ such that $\langle a, b\rangle$ is soluble for all $a \in A$ a…
The nonsoluble length $λ(G)$ of a finite group $G$ is defined as the number of nonsoluble factors in a shortest normal series each of whose factors either is soluble or is a direct product of nonabelian simple groups. The generalized Fitting height of a finite group $G$ is the least number $h=h^*(G)$ such that $F^*_h(G)=G$, where $F^*_1(G)=F^*(G)$ is the generalized Fitting subgroup, and $F^*_{i+1}(G)$ is the inverse image of $F^*(G/F^*_{i}(G))$. It is proved that if a finite group $G=AB$ is factorized by two subgroups of coprime orders, then the nonsoluble length of $G$ is bounded in terms of…
On the Length of a Maximal Subgroup of a Finite Group
本文证明有限群极大子群的广义Fitting高度和非-p-可解长度有下界，并通过构造遗传饱和群系反例，指出两篇已发表论文中相关结果有误。
The generalized Fitting height $h^*(G)$ of a finite group $G$ is the least number $h$ such that $\mathrm{F}_h^* (G) = G$, where $\mathrm{F}_{(0)}^* (G) = 1$, and $\mathrm{F}_{(i+1)}^*(G)$ is the inverse image of the generalized Fitting subgroup $\mathrm{F}^*(G/\mathrm{F}^*_{(i)} (G))$. Let $p$ be a prime, $1=G_0\leq G_1\leq\dots\leq G_{2h+1}=G$ be the shortest normal series in which for $i$ odd the factor $G_{i+1}/G_i$ is $p$-soluble (possibly trivial), and for $i$ even the factor $G_{i+1}/G_i$ is a (non-empty) direct product of nonabelian simple groups. Then $h=λ_p(G)$ is called the non-$p$-s…
Length parameters of finite groups and their Hall subgroups
研究有限群及其Hall子群的长度参数，证明若有限群G有Hallπ-子群H，则G的非p-可解长度受H的广义Fitting高度限制。
Some properties of abnormal subgroups in generalized soluble groups will be considered. In particular, the transitivity of abnormality in metahypercentral groups is proven. Also it will be proven that a subgroup H of a radical group G is abnormal in G if and only if every intermediate subgroup for H coincides with its normalizer in G. This result will extend on radical groups the well-known criterion of abnormality for finite soluble groups obtained by D. Taunt. For some infinite groups (not only periodic) the existence of Carter subgroups and their conjugations will be also proven.
The nonsoluble length $λ(G)$ of a finite group $G$ is defined as the minimum number of nonsoluble factors in a normal series of $G$ each of whose quotients either is soluble or is a direct product of nonabelian simple groups. The generalized Fitting height of a finite group $G$ is the least number $h=h^*(G)$ such that $F^*_h(G)=G$, where $F^*_1(G)=F^*(G)$ is the generalized Fitting subgroup, and $F^*_{i+1}(G)$ is the inverse image of $F^*(G/F^*_{i}(G))$. In the present paper we prove that if $λ(J)\leq k$ for every 2-generator subgroup $J$ of $G$, then $λ(G)\leq k$. It is conjectured that if $h…
Let $G$ be some generalized $π$-soluble groups and ${\cal F}$ be a Fitting set of $G$. In this paper, we prove the existence and conjugacy of ${\cal F}$-injectors of $G$, and give a description of the structure of the injectors.
In this paper the concept of $\mathbb{F}$-functorial of a finite group was introduced. These functorials have many properties of the Fitting subgroup of a soluble group and the generalized Fitting subgroup of a finite group. It was shown that the set of all $\mathbb{F}$-functorials is a complete distributive lattice and the cardinality of this lattice is continuum. The sharp bounds on the generalized Fitting height of mutually permutable product of two subgroups were obtained.
The generalized Fitting height of a finite group $G$ is the least number $h=h^*(G)$ such that $F^*_h(G)=G$, where the $F^*_i(G)$ is the generalized Fitting series: $F^*_1(G)=F^*(G)$ and $F^*_{i+1}(G)$ is the inverse image of $F^*(G/F^*_{i}(G))$. It is proved that if $G$ admits a soluble group of automorphisms $A$ of coprime order, then $h^*(G)$ is bounded in terms of $h^* (C_G(A))$, where $C_G(A)$ is the fixed-point subgroup, and the number of prime factors of $|A|$ counting multiplicities. The result follows from the special case when $A=\langleφ\rangle$ is of prime order, where it is proved …
We consider the class $\mathfrak M$ of $\bf R$--modules where $\bf R$ is an associative ring. Let $A$ be a module over a group ring $\bf R$$G$ where $G$ is a group and let $\mathfrak L(G)$ be a set of all proper subgroups of $G$ such that if $H \in \mathfrak L(G)$ then $A/C_{A}(H)$ belongs to $\mathfrak M$. We study an $\bf R$$G$--module $A$ such that $G \not = G'$, $C_{G}(A) = 1$, $A/C_{A}(G) \not \in \mathfrak M$, and $\mathfrak M$ is one of the classes: artinian $\bf R$--modules, minimax $\bf R$--modules, finite $\bf R$--modules. We consider the cases: 1) $\mathfrak M$ is a class of all art…
Let $g$ be an element of a finite group $G$. For a positive integer $n$, let $E_n(g)$ be the subgroup generated by all commutators $[...[[x,g],g],\dots,g]$ over $x\in G$, where $g$ is repeated $n$ times. By Baer's theorem, if $E_n(g)=1$, then $g$ belongs to the Fitting subgroup $F(G)$. We generalize this theorem in terms of certain length parameters of $E_n(g)$. For soluble $G$ we prove that if, for some $n$, the Fitting height of $E_n(g)$ is equal to $k$, then $g$ belongs to the $(k+1)$th Fitting subgroup $F_{k+1}(G)$. For nonsoluble $G$ the results are in terms of nonsoluble length and gener…
A contribution to the theory of $σ$-properties of a finite group
本文通过引入正规子群D避开主因子的条件，分别给出了有限群超可解性与有限可解群为PST-群的充要刻画，推进了有限群σ-性质理论。
Coverings and liftings of generalized crossed modules
本文定义广义cat$^1$-群并构造其到广义交叉模的函子，引入覆盖与提升概念，旨在证明给定广义交叉模的覆盖范畴与提升范畴等价。
We show that two blocks of generalized quaternion defect with three simple modules over a sufficiently large $2$-adic ring $\mathcal O$ are Morita-equivalent if and only if the corresponding blocks over the residue field of $\mathcal O$ are Morita-equivalent. As a corollary we show that any two blocks defined over $\mathcal O$ with three simple modules and the same generalized quaternion defect group are derived equivalent.
In math.GR/0510298, we showed that every F-quasigroup is linear over a special kind of Moufang loop called an NK-loop. Here we extend this relationship by showing an equivalence between the equational class of (pointed) F-quasigroups and the equational class corresponding to a certain notion of generalized module (with noncommutative, nonassociative addition) for an associative ring.
On commutative invariants for modules over crossed products of minimax nilpotent linear groups
本文针对有限秩可解群$G$的极小极大幂零无挠正规子群$N$与有限生成交换整环$R$的交叉积$R*N$上的模$W$，构建其与特定素理想等价类集$M$的对应关系，证明$G$对$N$的共轭作用可延拓至$M$，为用交换代数方法研究$W$提供基础。
On the generalized Bykovskii presentation of Steinberg modules
本研究将整数环Steinberg模的Bykovskii表示推广至高斯整数与艾森斯坦整数情形，同时证明该推广不适用于若干欧几里得数环的Steinberg模，推进了数环特殊线性群虚拟对偶模表示理论的研究。
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