Vdovin Evgenii
In the paper we complete the classification of Carter subgroups in finite almost simple groups. In particular, we prove that Carter subgroups of every finite almost simple group are conjugate. Togeather with previous results by author and F. Dalla Volta, A. Lucchini, and M. C. Tamburini, as a corollary, it follows that Carter subgroups of every finite group are conjugate.
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In the paper it is proven that Carter subgroups of a finite group are conjugate. A complete classification of Carter subgroups in finite almost simple groups is also obtained.
It is proven in the paper, that Carter subgroups of a finite group are conjugate if Carter subgroups in the group of induced automorphisms for every non-Abelian composition factor are conjugate.
We show that a finite permutation group containing a regular abelian self-normalizing subgroup is soluble.
In the note we prove that all composition factors of a finite group possessing a Carter subgroup of odd order either are abelain, or are isomorphic to $L_2(3^{2n+1})$.
Let $π$ be a set of primes. We show that $π$-separable groups have a conjugacy class of subgroups which specialize to Carter subgroups, i.e. self-normalizing nilpotent subgroups, or equivalently, nilpotent projectors, when specializing to soluble groups.
In the paper we obtain the existence criterion of a Carter subgroup in a finite group in terms of its normal series. An example showing that the criterion cannot be reformulated in terms of composition factors is given.
We consider an amalgam of groups constructed from fusion systems for different odd primes p and q. This amalgam contains a self-normalizing cyclic subgroup of order pq and isolated elements of order p and q.
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.
Counting in nilpotent injectors and Carter subgroups
本文考察有限可解群及 ${\mathcal N}$-约束群中，包含给定子群的幂零内射子与幂零投影子族的数论计数性质。
Let $G$ be a finite solvable group and $H$ be a subgroup of $Aut(G)$. Suppose that there exists an $H$-invariant Carter subgroup $F$ of $G$ such that the semidirect product $FH$ is a Frobenius group with kernel $F$. We prove that the terms of the Fitting series of $C_{G}(H)$ are obtained as the intersection of $C_{G}(H)$ with the corresponding terms of the Fitting series of $G$, and the Fitting height of $G$ may exceed the Fitting height of $C_{G}(H)$ by at most one. As a corollary it is shown that for any set of primes $π$, the terms of the $π$-series of $C_{G}(H)$ is obtained as the intersec…
We modify the Frecon-Jaligot construction of Carter subgroups to show that a degenerate type group has a Carter subgroup invariant under the Sylow 2-subgroup of a group of automorphisms; thus reducing the need to know that Carter subgroups are conjugate in degenerate type groups.
This paper provides new, relatively simple proofs of some important results about unipotent classes in simple linear algebraic groups. We derive the formula for the Jordan blocks of the Richardson class of a parabolic subgroup of a classical group. This result was originally due to Spaltenstein. Secondly, we derive, for good characteristic, the description of the natural partial order of unipotent classes of a classical group in terms of their Jordan blocks. This result was originally due to Gerstenhaber and Hesselink. As a consequence we obtain a proof of the Bala--Carter Theorem which holds …
Carter-like theorems for skew braces
本文引入 Carter 子斜辫子，分析其结构限制，并证明在特定条件下有限可解斜辫子及有限多重置换斜辫子中存在此类子结构。
This paper describes how to use subgroups to parameterize unipotent classes in the classical algebraic group in characteristic 2. These results can be viewed as an extension of the Bala-Carter Theorem, and give a convenient way to compare unipotent classes in a group $G$ with unipotent classes of a subgroup $X$ where $G$ is exceptional and $X$ is a Levi subgroup of classical type.
We present unpublished work of D.Carter, G.Keller, and E.Paige on bounded generation in special linear groups. Let n be a positive integer, and let A = O be the ring of integers of an algebraic number field K (or, more generally, let A be a localization O_S.) If n = 2, assume that A has infinitely many units. We show there is a finite-index subgroup H of SL(n,A), such that every matrix in H is a product of a bounded number of elementary matrices. We also show that if T is in SL(n,A), and T is not a scalar matrix, then there is a finite-index, normal subgroup N of SL(n,A), such that every eleme…
A finite group $P$ is said to be \emph{primary} if $|P|=p^{a}$ for some prime $p$. We say a primary subgroup $P$ of a finite group $G$ satisfies the \emph{Frobenius normalizer condition} in $G$ if $N_{G}(P)/C_{G}(P)$ is a $p$-group provided $P$ is $p$-group. In this paper, we determine the structure of a finite group $G$ in which every non-subnormal primary subgroup satisfies the Frobenius normalized condition. In particular, we prove that if every non-normal primary subgroup of $G$ satisfies the Frobenius condition, then $G/F(G)$ is cyclic and every maximal non-normal nilpotent subgroup $U$ o…
We consider factorizations of a finite group $G$ into conjugate subgroups, $G=A^{x_{1}}\cdots A^{x_{k}}$ for $A\leq G$ and $x_{1},\ldots ,x_{k}\in G$, where $A$ is nilpotent or solvable. First we exploit the split $BN$-pair structure of finite simple groups of Lie type to give a unified self-contained proof that every such group is a product of four or three unipotent Sylow subgroups. Then we derive an upper bound on the minimal length of a solvable conjugate factorization of a general finite group. Finally, using conjugate factorizations of a general finite solvable group by any of its Carter…
Dehn functions of coabelian subgroups of direct products of groups
本研究提出计算群直积的余阿贝尔子群精确Dehn函数的新方法，推广了前人相关结果，并将其应用于多类群的相关研究中。
One of the central tools in the classification of simple algebraic groups is the distinction between semisimple subgroups and unipotent subgroups. It is not a priori clear how to make this distinction for torsion-free subgroups of a group of finite Morley rank. We exploit the ``graded'' notion of 0-unipotence to develop a Sylow theory for torsion-free subgroups of a solvable group of finite Morley rank. This Sylow theory provides a robust alternative to the usual theory of Carter subgroups, and will be used in the analysis of intersections of Borel subgroups in minimal simple groups.
We announce an atlas of subgroup lattices of almost simple groups and present two algorithms that were used to produce the atlas.
We construct new examples of CAT(0) groups containing non finitely presented subgroups that are of type $FP_2$, these CAT(0) groups do not contain copies of $\mathbb{Z}^3$. We also give a construction of groups which are of type $F_n$ but not $F_{n+1}$ with no free abelian subgroups of rank greater than $\lceil\frac{n}{3}\rceil$.
Above Isaacs' Head Characters
本文研究了正常可解子群的伊萨克斯头部特征标，并给出了有限群共轭类数的新下界。
In this paper we provide a sufficient condition for a subgroup of a free group not to have the almost congruence extension property (ACEP). We also show that any finitely generated subgroup of a free group satisfies some a generalization of ACEP.
We determine all maximal subgroups of the almost simple groups with socle $T=\PSL(2,q)$, that is, of all groups $G$ such that $\PSL(2,q)\leqslant G\leqslant\PGammaL(2,q)$, with $q\geq 4$.
We prove that the tree almost automorphism groups admit exactly three commensurability classes of closed commensurated subgroups. Our proof utilizes an independently interesting characterization of subgroups of the tree almost automorphism groups which contain only periodic elements in terms of the dynamics of the action on the boundary of the tree. Our results further cover several interesting finitely generated subgroups of the tree almost automorphism groups, including the Thompson groups $F$, $T$, and $V$. We show in particular that Thompson's group $T$ has no commensurated subgroups other…
Topological Properties of Almost Abelian Groups
本文研究复殆阿贝尔李群的离散子群、紧性条件与极大紧子群等拓扑性质。
Characters and the Generation of Sylow 3-Subgroups For Almost Simple Groups
本研究针对几乎单群，证明其特征标表可算法判定Sylow 3-子群是否2生成，且该性质等价于主块特征标的Galois作用条件，是Alperin-McKay-Navarro猜想的推论。
We show that every countable group embeds in a group of type $FP_2$.
A subgroup $H\leq G$ is said to be almost normal if every conjugate of $H$ is commensurable to $H$. If $H$ is almost normal, there is a well-defined quotient space $G/H$. We show that if a group $G$ has type $F_{n+1}$ and contains an almost normal coarse $PD_n$ subgroup $H$ with $e(G/H)=\infty$, then whenever $G'$ is quasi-isometric to $G$, it contains an almost normal subgroup $H'$ that is quasi-isometric to $H$. Moreover, the quotient spaces $G/H$ and $G'/H'$ are quasi-isometric. This generalises a theorem of Mosher-Sageev-Whyte, who prove the case in which $G/H$ is quasi-isometric to a fini…
On the maximal subgroups of almost simple and primitive perfect groups
本文证明有限几乎单群的最大子群的导出列第10项是完全群，并推广到完全群的无核最大子群，且常数10是最优的。
For a finite group $G$, let $d(G)$ denote the minimal number of elements required to generate $G$. In this paper, given a finite almost simple group $G$ and any maximal subgroup $H$ of $G$, we determine a precise upper bound for $d(H)$. In particular, we show that $d(H)\leq 5$, and that $d(H)\geq 4$ if and only if $H$ occurs in a known list. This improves a result of Burness, Liebeck and Shalev. The method involves the theory of crowns in finite groups.
Large maximal subgroups of almost simple classical groups
本文刻画以有限单经典群为基座的几乎单群的大极大子群，确定了相关几乎单群集合非空的充要条件。
An almost Abelian Lie group is a non-Abelian Lie group with a codimension 1 Abelian normal subgroup. The majority of 3-dimensional real Lie groups are almost Abelian, and they appear in all parts of physics that deal with anisotropic media - cosmology, crystallography etc. In theoretical physics and differential geometry, almost Abelian Lie groups and their homogeneous spaces provide some of the simplest solvmanifolds on which a variety of geometric structures such as symplectic, Kähler, spin etc., are currently studied in explicit terms. Recently, almost Abelian Lie algebras were classified a…
We prove in this paper that a finite almost simple group $R$ with socle the non-abelian simple group $S$ possesses a conjugacy class of core-free maximal subgroups whose index coincides with the smallest index $\operatorname{l}(S)$ of a maximal group of $S$ or a conjugacy class of core-free maximal subgroups with a fixed index $v_S \leq {\operatorname{l}(S)^2}$, depending only on $S$. We show that the number of subgroups of the outer automorphism group of $S$ is bounded by $\log^3 {\operatorname{l}(S)}$ and $\operatorname{l}(S)^2 < |S|$.
On the Maximal Subgroups of $E_7(q)$ and Related Almost Simple Groups
本文几乎完全分类了 $E_7(q)$ 的极大子群，仅缺失四个特殊情形，并完整确定了 $q$ 为 $3$ 的幂及 $q=4$ 时的极大子群。
Given an infinite topological group G and a cardinal k>0, we say that G is almost k-free if the set of k-tuples in G^k which freely generate free subgroups of G is dense in G^k. In this note we examine groups having this property and construct examples. For instance, we show that if G is a non-discrete Hausdorff topological group that contains a dense free subgroup of rank k>0, then G is almost k-free. A consequence of this is that for any infinite set X, the group of all permutations of X is almost 2^|X|-free. We also show that an infinite topological group is almost aleph_0-free if and only …
Finite groups whose maximal subgroups have almost odd index
本文研究了由群不变量的算术性质决定有限群结构的问题，确定了极大子群的指数不被4整除的有限群的结构，并由此得到了一些新的2-幂零性准则。
The maximal subgroups of the exceptional groups $F_4(q)$, $E_6(q)$ and ${}^2E_6(q)$ and related almost simple groups
本文综合代数群、表示论、计算代数与三线性型方法，完整确定所有有限域上 $F_4(q)$、$E_6(q)$、${}^2E_6(q)$ 及相关几乎单群的极大子群，并更正 ${}^2F_4(8)$ 的列表。
A subgroup of a finite group G is said to be second maximal if it is maximal in every maximal subgroup of G that contains it. A question which has received considerable attention asks: can every positive integer occur as the number of the maximal subgroups that contain a given second maximal subgroup in some finite group G? Various reduction arguments are available except when G is almost simple. Following the classification of the finite simple groups, finite almost simple groups fall into three categories: alternating and symmetric groups, almost simple groups of Lie type, sporadic groups an…
Indeed, the Monster has no almost simple maximal subgroup with socle $\text{PSL}_2(16)$
该研究证明了Monster群不存在基柱为$\text{PSL}_2(16)$的几乎单极大子群，验证了Holmes与威尔逊未发表结果中的对应情形，证明基于mmgroup包的可复现计算，剩余两种情形将在扩展版中处理。
For a prime $p$, a $p$-subgroup of a finite group $G$ is said to be large if and only if $Q= F^*(N_G(Q))$ and, for all $1 \neq U \le Z(Q)$, $N_G(U) \le N_G(Q)$. In this article we determine those groups $G$ which have a large subgroup and which in addition have a proper subgroup $H$ containing a Sylow $p$-subgroup of $G$ with $F^*(H)$ a group of Lie type in characteristic $p$ and rank at least 2 (excluding $\PSL_3(p^a)$) and $C_H(z)$ soluble for some $z \in Z(S)$. This work is part of a project to determine the groups $G$ which contain a large $p$-subgroup.
We derive presentations of the interval groups related to all quasi-Coxeter elements in the Coxeter group of type $D_n$. Type $D_n$ is the only infinite family of finite Coxeter groups that admits proper quasi-Coxeter elements. The presentations we obtain are over a set of generators in bijection with what we call a Carter generating set, and the relations are those defined by the related Carter diagram together with a twisted or a cycle commutator relator, depending on whether the quasi-Coxeter element is a Coxeter element or not. The proof is based on the description of two combinatorial tec…
Let $G$ be a finite almost simple group with socle $G_0$. A (nontrivial) factorization of $G$ is an expression of the form $G=HK$, where the factors $H$ and $K$ are core-free subgroups. There is an extensive literature on factorizations of almost simple groups, with important applications in permutation group theory and algebraic graph theory. In a recent paper, Li and Xia describe the factorizations of almost simple groups with a solvable factor $H$. Several infinite families arise in the context of classical groups and in each case a solvable subgroup of $G_0$ containing $H \cap G_0$ is iden…
An invariant random subgroup $H \leq G$ is a random closed subgroup whose law is invariant to conjugation by all elements of $G$. When $G$ is locally compact and second countable, we show that for every invariant random subgroup $H \leq G$ there almost surely exists an invariant measure on $G/H$. Equivalently, the modular function of $H$ is almost surely equal to the modular function of $G$, restricted to $H$. We use this result to construct invariant measures on orbit equivalence relations of measure preserving actions. Additionally, we prove a mass transport principle for discrete or compact…
The structure and generation of the second maximal subgroups of the almost simple groups with alternating, classical or sporadic socle
针对基座为交错、古典或零散群的几乎单群，修正了其非抛物极大子群生成元数量的分类，将生成元界优化至最优值，确定了相关主因子并改进了已有结论。
Maximally almost periodic subgroups of elementary Abelian topological groups
本文研究了无限Abelian拓扑群的极大几乎周期子群，证明了任何无限Abelian拓扑群的素数指数都具有无限的极大几乎周期子群。
The study of finite subgroups of a simple algebraic group $G$ reduces in a sense to those which are almost simple. If an almost simple subgroup of $G$ has a socle which is not isomorphic to a group of Lie type in the underlying characteristic of $G$, then the subgroup is called non-generic. This paper considers non-generic subgroups of simple algebraic groups of exceptional type in arbitrary characteristic. A finite subgroup is called Lie primitive if it lies in no proper subgroup of positive dimension. We prove here that many non-generic subgroup types, including the alternating and symmetric…
In this paper we examine embeddings of alternating groups and symmetric groups into almost simple groups of exceptional type. In particular, we prove that unless the alternating or symmetric group has degree 6 or 7, there is no maximal subgroup of any almost simple group with socle an exceptional group of Lie type that is an alternating or symmetric group. Furthermore, in the remaining open cases we give considerable information about the possible embeddings. Note that no maximal alternating or symmetric subgroups are known in the remaining cases. This is the first in a sequence of papers aimi…
We study embeddings of $\mathrm{PSL}_2(p^a)$ into exceptional groups $G(p^b)$ for $G=F_4,E_6,{}^2\!E_6,E_7$, and $p$ a prime with $a,b$ positive integers. With a few possible exceptions, we prove that any almost simple group with socle $\mathrm{PSL}_2(p^a)$, that is maximal inside an almost simple exceptional group of Lie type $F_4$, $E_6$, ${}^2\!E_6$ and $E_7$, is the fixed points under the Frobenius map of a corresponding maximal closed subgroup of type $A_1$ inside the algebraic group. Together with a recent result of Burness and Testerman for $p$ the Coxeter number plus one, this proves t…
Assume that a finite almost simple group with simple socle isomorphic to an exceptional group of Lie type possesses a solvable Hall subgroup. Then there exist four conjugates of the subgroup such that their intersection is trivial.
In this paper, we classify finite quasiprimitive permutation groups with a metacyclic transitive subgroup, solving a problem initiated by Wielandt in 1949. It also involves the classification of factorizations of almost simple groups with a metacyclic factor.
The aim of this brief note is to offer another proof of a theorem of Hrushovski that approximate subgroups of GL_n(C) are almost nilpotent. This approach generalizes to uniformly non amenable groups.
We consider the following problem stated by Vdovin (2010) in the "Kourovka notebook" (Problem 17.41): Let $H$ be a solvable subgroup of a finite group $G$ that has no nontrivial solvable normal subgroups. Do there always exist five conjugates of $H$ whose intersection is trivial? This problem is closely related to a conjecture by Babai, Goodman and Pyber (1997) about an upper bound for the index of a normal solvable subgroup in a finite group. In particular, a positive answer to Vdovin's problem yields that if $G$ has a solvable subgroup of index $n$, then it has a solvable normal subgroup of …
We consider the isomorphism problem for almost split Kac--Moody groups, which have been constructed by Rémy via Galois descent from split Kac-Moody groups as defined by Tits. We show that under certain technical assumptions, any isomorphism between two such groups must preserve the canonical subgroup structure, i.e. the twin root datum associated to these groups, which generalizes results of Caprace in the split case. An important technical tool we use is the existence of maximal split subgroups inside almost split Kac-Moody groups, which generalizes the corresponding result of Borel-Tits for …
In this paper, we study the action of finite subgroups of the mapping class group of a surface on the curve complex. We prove that if the diameter of the almost fixed point set of a finite subgroup H is big enough, then the centralizer of H is infinite.
When are Two Subgroups Independent?
本文提出子群独立的新定义，指出传统几乎不相交条件不足，给出了必要和充分条件，并提出算法用于判断子群独立性。
In this paper we create many examples of hyperbolic groups with subgroups satisfying interesting finiteness properties. We give the first examples of subgroups of hyperbolic groups which are of type $FP_2$ but not finitely presented. We give uncountably many groups of type $FP_2$ with similar properties to those subgroups of hyperbolic groups. Along the way we create more subgroups of hyperbolic groups which are finitely presented but not of type $FP_3$.
It is shown that for a solvable subgroup $G$ of an almost simple group $S$ which socle is isomorphic to $A_n$ $ (n\ge5)$ there are $x,y,z,t \in S$ such that $G \cap G^x \cap G^y \cap G^z \cap G^t =1.$
In this paper we find the number of conjugate $π$-Hall subgroups in all finite almost simple groups. We also complete the classification of $π$-Hall subgroups in finite simple groups and correct some mistakes from our previous paper.
The maximal subgroups of the Monster
本文利用Python包mmgroup的可复现计算，解决了魔群极大子群分类的遗留问题，发现两个新极大子群，纠正了此前的错误论断，完成了全部散在单群的极大子群分类。
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