S. P. Glasby
Let c(d) denote the minimal composition length of all finite solvable groups with solvable (or derived) length d. We prove that c(d) equals 0,1,2,4,5,7,8,13,15 when d=0,1,2,3,4,5,6,7,8.
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It has been shown by several authors that there exists a non-solvable Bieberbach group of dimension $15$. In this note we show that this is in fact a minimal dimension for such kind of groups.
We investigate properties of finite transitive permutation groups $(G, Ω)$ in which all proper subgroups of $G$ act intransitively on $Ω.$ In particular, we are interested in reduction theorems for minimally transitive representations of solvable groups.
A group is called a CA-group if the centralizer of every non-central element is abelian. Furthermore, a group is called a minimal non-CA-group if it is not a CA-group itself, but all of its proper subgroups are. In this paper, we give a classification of the finite non-solvable minimal non-CA-groups.
We prove that the minimal composition length, $c$, of a solvable group with solvable length $d$ satisfies $9^{(d-3)/9}< c< 9^{(d+1)/5}$, and the minimal composition length, $c^o$, of a group with odd order and solvable length $d$ satisfies $7^{(d-2)/5}< c^o< 2^d$.
Let $V$ be a finite vector space over a finite field of order $q$ and of characteristic $p$. Let $G\leq GL(V)$ be a $p$-solvable completely reducible linear group. Then there exists a base for $G$ on $V$ of size at most $2$ unless $q \leq 4$ in which case there exists a base of size at most $3$. The first statement extends a recent result of Halasi and Podoski and the second statement generalizes a theorem of Seress. An extension of a theorem of Pálfy and Wolf is also given.
In this paper we completely characterize solvable real Lie groups definable in o-minimal expansions of the real field.
Let π(G) denote the set of prime divisors of the order of a finite group G. The prime graph of G is the graph with vertex set π(G) with edges {p,q} if and only if there exists an element of order pq in G. In this paper, we prove that a graph is isomorphic to the prime graph of a solvable group if and only if its complement is 3-colorable and triangle free. We then introduce the idea of a minimal prime graph. We prove that there exists an infinite class of solvable groups whose prime graphs are minimal. We prove the 3k-conjecture on prime divisors in element orders for solvable groups with mini…
We prove that the base size of a transitive group $G$ with solvable point stabilizer and with trivial solvable radical is not greater than $k$ provided the same statement holds for the group of $G$-induced automorphisms of each nonabelian composition factor of $G$.
We explore graph theoretical properties of minimal prime graphs of finite solvable groups. In finite group theory studying the prime graph of a group has been an important topic for the past almost half century. Recently prime graphs of solvable groups have been characterized in graph theoretical terms only. This now allows the study of these graphs with methods from graph theory only. Minimal prime graphs turn out to be of particular interest, and in this paper we pursue this further by exploring, among other things, diameters, Hamiltonian cycles and the property of being self-complementary f…
Kernels of minimal characters of solvable groups
本文研究有限可解群中极小次数非线性不可约特征标，证明当其次数为奇数时，相应特征标核所得商群具有幂零-by-阿贝尔结构。
Let $PL_0(I)$ represent the group of orientation-preserving piecewise-linear homeomorphisms of the unit interval which admit finitely many breaks in slope, under the operation of composition. We find a non-solvable group $W$ and show that $W$ embeds in every non-solvable subgroup of $PL_0(I)$. We find mild conditions under which other non-solvable subgroups ($B$, $(\wr\Z\wr)^{\infty}$, $(\Z\wr)^{\infty}$, and $^{\infty}(\wr\Z)$) embed in subgroups of $PL_0(I)$. We show that all solvable subgroups of $PL_0(I)$ embed in all non-solvable subgroups of $PL_0(I)$. These results continue to apply if …
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…
Nilpotent groups, solvable groups and factorizable inverse monoids
本文引入逆半群中的次中心与中心幂等元序列，证明了可因子化逆幺半群中此类序列的同构性（类比Jordan-Hölder定理），并据此定义了G-幂零与G-可解逆幺半群，为研究幂零群和可解群提供了新途径。
In this paper, we study the product of orders of composition factors of odd order in a composition series of a finite linear group. First we generalize a result by Manz and Wolf about the order of solvable linear groups of odd order. Then we use this result to find bounds for the product of orders of composition factors of odd order in a composition series of a finite linear group.
A. Smoktunowicz and L. Vendramin conjectured that if $A$ is a finite skew brace with solvable additive group, then the multiplicative group of $A$ is solvable. In this short note we make a step towards positive solution of this conjecture proving that if $A$ is a minimal finite skew brace with solvable additive group and non-solvable multiplicative group, then the multiplicative group of $A$ is not simple. On the way to obtaining this result, we prove that the conjecture of A. Smoktunowicz and L. Vendramin is correct in the case when the order of $A$ is not divisible by $3$.
For the class of solvable groups of homeomorphisms of the line preserving orientation and containing a freely acting element, we establish the metabelianity of the quotient group $G/H_G$, where the elements of the normal subgroup $H_G$ are stabilizers of the minimal set. This fact is an important element in the classification theorem, used, in particular, in the study of the Thompson's group $F$.
It is shown that certain ascending HNN extensions of free abelian groups of finite rank, as well as various lamplighter groups, can be realized as automaton groups, i.e., can be given a self-similar structure. This includes the solvable Baumslag-Solitar groups BS(1,m), for m not +/-1. In addition, it is shown that, for any relatively prime integers m,n>=2, the pair of Baumslag-Solitar groups BS(1,m) and BS(1,n) can be realized by a pair of dual automata. The examples are then used to illustrate more general connections between Schreier graphs, composition of automata and dual automata.
On the cut-set of the Gruenberg-Kegel graph of a finite solvable group
本文证明有限可解群的Gruenberg-Kegel图割集对应长度为5的受控σ-列，给出割点情形Fitting长度的最优上界，并刻画具有二元极小割集的图结构。
Quadratic fields of values of finite solvable groups
该研究针对有限可解群G，在其特征值生成的扩域E为二次域的条件下，确定了E的结构，补充了含阶7合成因子的情况，部分解答了G. Navarro的问题。
The prime graph $Γ(G)$ of a finite group $G$ (also known as the Gruenberg-Kegel graph) has as its vertices the prime divisors of $|G|$, and $p\text-q$ is an edge in $Γ(G)$ if and only if $G$ has an element of order $pq$. Since their inception in the 1970s these graphs have been studied extensively; however, completely classifying the possible prime graphs for larger families of groups remains a difficult problem. For solvable groups such a classification was found in 2015. In this paper we go beyond solvable groups for the first time and characterize prime graphs of a more general class of gro…
We study topological groups having all closed subgroups (totally) minimal and we call such groups c-(totally) minimal. We show that a locally compact c-minimal connected group is compact. Using a well-known theorem of Hall and Kulatilaka and a characterization of a certain class of Lie groups, due to Grosser and Herfort, we prove that a c-minimal locally solvable Lie group is compact. It is shown that if a topological group $G$ contains a compact open normal subgroup $N$, then $G$ is c-totally minimal if and only if $G/N$ is hereditarily non-topologizable. Moreover, a c-totally minimal group t…
Solvable Supplements to Normalizers of Cyclic 2-Subgroups
本文放宽了可解补充条件至仅对循环2-子群成立，刻画了满足该条件的有限群的非交换合成因子为特定PSL2(q)族，并给出可解性的充要条件。
M.Newman has asked if it is the case that whenever H and K are isomorphic subgroups of a finite solvable group G with H maximal, then K is also maximal. This question was considered in a paper of I.M. Isaacs and the second author, where (among other things) the answer was shown to be affirmative if H has an Abelian Sylow 2-subgroup. Here, we show that the answer is affirmative unless the index of H is a power of a prime less than 5 and we obtain further restrictions on the structure of a purported minimal counterexample.
Let $G$ be a finite group, $μ$ be the Möbius function on the subgroup lattice of $G$, and $λ$ be the Möbius function on the poset of conjugacy classes of subgroups of $G$. It was proved by Pahlings that, whenever $G$ is solvable, the property $μ(H,G)=[N_{G^\prime}(H):G^{\prime}\cap H]\cdotλ(H,G)$ holds for any subgroup $H$ of $G$. It is known that this property does not hold in general; for instance it does not hold for every simple groups, the Mathieu group $M_{12}$ being a counterexample. In this paper we investigate the relation between $μ$ and $λ$ for some classes of non-solvable groups; a…
Prime Graphs of Infinite Groups
本文定义无限群的素图，并将有限可解及T-可解群的素图结果推广至相应无限群类别。
Realizing Gruenberg-Kegel graphs of $T$-solvable groups with structurally simplified extensions of $T$
本文研究了T-可解群的素数图补集，证明其可由可解群和准简单T-可解群作用于元素阿贝尔群的直积实现，并探讨了PSL(2,13)-可解群的素数图补集分类。
2-covering numbers of some finite solvable groups
本文推翻了关于有限可解群2覆盖数的猜想，指出其不满足σ₂(G)=1+q+q²的结论。
On degrees of minimal invariant characters
本文证明具有恰好两个最小G-不变特征次数的正规子群是可解的，并探讨了次数为{1, f}时f为素幂或换位子群交换的条件。
We say that a group is anti-solvable if all of its composition factors are non-abelian. We consider a particular family of anti-solvable finite groups containing the simple alternating groups for $n\neq 6$ and all 26 sporadic simple groups. We prove that, if $K$ is a perfect field and $X$ is a homogeneous space of a smooth algebraic $K$-group $G$ with finite geometric stabilizers lying in this family, then $X$ is dominated by a $G$-torsor. In particular, if $G=\mathrm{SL}_n$, all such homogeneous spaces have rational points.
A covering of a group is a finite set of proper subgroups whose union is the whole group. A covering is minimal if there is no covering of smaller cardinality, and it is nilpotent if all its members are nilpotent subgroups. We complete a proof that every group that has a nilpotent minimal covering is solvable, starting from the previously known result that a minimal counterexample is an almost simple finite group.
We prove that the rank (that is, the minimal size of a generating set) of lattices in a general connected Lie group is bounded by the co-volume of the projection of the lattice to the semi-simple part of the group. This was proved by Gelander for semi-simple Lie groups and by Mostow for solvable Lie groups. Here we consider the general case, relying on the semi-simple case. In particular, we extend Mostow's theorem from solvable to amenable groups.
Let $G$ be a finite group. Denoting by ${\rm{cd}}(G)$ the set of the degrees of the irreducible complex characters of $G$, we consider the {\it character degree graph} of $G$: this is the (simple, undirected) graph whose vertices are the prime divisors of the numbers in ${\rm{cd}}(G)$, and two distinct vertices $p$, $q$ are adjacent if and only if $pq$ divides some number in ${\rm{cd}}(G)$. This paper completes the classification, started in [5] and [6], of the finite non-solvable groups whose character degree graph has a {\it cut-vertex}, i.e. a vertex whose removal increases the number of co…
Let $G$ be a finite group, and let ${\rm{cd}}(G)$ denote the set of degrees of the irreducible complex characters of $G$. Define then the character degree graph $Δ(G)$ as the (simple undirected) graph whose vertices are the prime divisors of the numbers in ${\rm{cd}}(G)$, and two distinct vertices $p$, $q$ are adjacent if and only if $pq$ divides some number in ${\rm{cd}}(G)$. This paper continues the work, started in [7], toward the classification of the finite non-solvable groups whose degree graph possesses a cut-vertex, i.e., a vertex whose removal increases the number of connected compone…
Let $G$ be a finite group and let $ram^{t}(G)$ denote the minimal positive integer $n$ such that $G$ can be realized as the Galois group of a tamely ramified extension of $\mathbb{Q}$ ramified only at $n$ finite primes. Let $d(G)$ denote the minimal non negative integer for which there exists a subset $X$ of $G$ with $d(G)$ elements such that the normal subgroup of $G$ generated by $X$ is all of $G$. It is known that $d(G)\leq ram^{t}(G)$. However, it is unknown whether or not every finite group $G$ can be realized as a Galois group of a tamely ramified extension of $\mathbb{Q}$ with exactly $…
Classifying prime graphs of finite groups -- a methodical approach
本文建立任意T-可解群素图边存在性的一般理论，并对阶含4个素因子的绝大多数T分类了T-可解群的素图，发现其补图几乎均为3-可着色且含少量三角形。
We produce two separate algebraic descriptions of the isomorphism classes of the solvable subgroups of the group PLo(I) of piecewise-linear orientation-preserving homeomorphisms of the unit interval under the operation of composition, and also of the solvable subgroups of the generalized R. Thompson groups F_n. The first description is as a set of isomorphism classes of groups which is closed under three algebraic operations, and the second is as the set of isomorphism classes of subgroups of a countable collection of easily described groups. We show the two descriptions are equivalent.
In the paper we consider the following conjecture: if a finite group $G$ possesses a solvable $π$-Hall subgroup $H$, then there exist elements $x,y,z,t\in G$ such that the identity $H\cap H^x\cap H^y\cap H^z\cap H^t=O_π(G)$ holds. The minimal counter example is shown to be an almost simple group of Lie type.
We investigate subgroups of the group PLo(I) of piecewise-linear, orientation-preserving homeomorphisms of the unit interval with finitely many breaks in slope, under the operation of composition, and also subgroups of the generalized Thompson groups F_n. We find geometric criteria determining the derived length of any such group, and use this criteria to produce a geometric classification of the solvable and non-solvable subgroups of PLo(I) and of the F_n. We also show that any standard restricted wreath product C wr T (of non-trivial groups) that embeds in PLo(I) or F_n must have T isomorphi…
On $p$-nonsingular systems of equations over solvable groups
本文研究可解群上 $p$-非奇异方程组的可解性，通过保持次正规序列性质的群嵌入，证明含不可解幺模方程的亚交换群最小阶为 42。
We continue the study of prime graphs of finite groups, also known as Gruenberg-Kegel graphs. The vertices of the prime graph of a finite group are the prime divisors of the group order, and two vertices $p$ and $q$ are connected by an edge if and only if there is an element of order $pq$ in the group. Prime graphs of solvable groups have been characterized in graph theoretical terms only, as have been the prime graphs of groups whose only nonsolvable composition factor is $A_5$. In this paper we classify the prime graphs of all groups whose composition factors have arithmetically small orders…
We study the nonabelian composition factors of a finite group $G$ assumed to admit an $\operatorname{Aut}(G)$-orbit of length at least $ρ|G|$, for a given $ρ\in\left(0,1\right]$. Our main results are the following: The orders of the nonabelian composition factors of $G$ are then bounded in terms of $ρ$, and if $ρ>\frac{18}{19}$, then $G$ is solvable. On the other hand, for each nonabelian finite simple group $S$, there is a constant $c(S)\in\left(0,1\right]$ such that $S$ occurs with arbitrarily large multiplicity as a composition factor in some finite group $G$ having an $\operatorname{Aut}(G…
On the finite solvable $PNC$-groups
本文研究有限可解PNC-群的基本性质，给出其为可解PNC-群的充分条件，并刻画最小非ON-群、ON-群及相关单群和子群条件。
In the paper we characterize the class of finite solvable groups by two-variable identities in a way similar to the characterization of finite nilpotent groups by Engel identities. More precisely, a sequence of words $u_1,...,u_n,... $ is called correct if $u_k\equiv 1$ in a group $G$ implies $u_m\equiv 1$ in a group $G$ for all $m>k$. We are looking for an explicit correct sequence of words $u_1(x,y),...,u_n(x,y),...$ such that a group $G$ is solvable if and only if for some $n$ the word $u_n$ is an identity in $G$. Let $u_1=x^{-2}y\min x$, and $u_{n+1} = [xu_nx\min,yu_ny\min]$. The main resu…
We study locally compact groups having all subgroups minimal. We call such groups hereditarily minimal. In 1972 Prodanov proved that the infinite hereditarily minimal compact abelian groups are precisely the groups $\mathbb Z_p$ of $p$-adic integers. We extend Prodanov's theorem to the non-abelian case at several levels. For infinite hypercentral (in particular, nilpotent) locally compact groups we show that the hereditarily minimal ones remain the same as in the abelian case. On the other hand, we classify completely the locally compact solvable hereditarily minimal groups, showing that in pa…
For a finite nonabelian group $G$ let $\rat(G)$ be the largest ratio of degrees of two nonlinear irreducible characters of $G$. We show that nonabelian composition factors of $G$ are controlled by $\rat(G)$ in some sense. Specifically, if $S$ different from the simple linear groups $\PSL_2(q)$ is a nonabelian composition factor of $G$, then the order of $S$ and the number of composition factors of $G$ isomorphic to $S$ are both bounded in terms of $\rat(G)$. Furthermore, when the groups $\PSL_2(q)$ are not composition factors of $G$, we prove that $|G:\Oinfty(G)|\leq \rat(G)^{21}$ where $\Oinf…
In this paper we continue the study of prime graphs of finite solvable groups. The prime graph, or Gruenberg-Kegel graph, of a finite group G has vertices consisting of the prime divisors of the order of G and an edge from primes p to q if and only if G contains an element of order pq. Since the discovery of a simple, purely graph theoretical characterization of the prime graphs of solvable groups in 2015 these graphs have been studied in more detail from a graph theoretic angle. In this paper we explore several new aspects of these graphs. We characterize regular reseminant graphs and study t…
The main purpose of this paper is to study the finite-dimensional solvable Lie algebras described in its title, which we call {\em minimal non-${\mathcal N}$}. To facilitate this we investigate solvable Lie algebras of nilpotent length $k$, and of nilpotent length $\leq k$, and {\em extreme} Lie algebras, which have the property that their nilpotent length is equal to the number of conjugacy classes of maximal subalgebras. We characterise the minimal non-${\mathcal N}$ Lie algebras in which every nilpotent subalgebra is abelian, and those of solvability index $\leq 3$.
We prove the definability, and actually the finiteness of the commutator width, of many commutator subgroups in groups definable in o-minimal structures. It applies in particular to derived series and to lower central series of solvable groups. Along the way, we prove some generalities on groups with the descending chain condition on definable subgroups and/or with a definable and additive dimension.
Criteria for Classifying Prime Graphs of PSL(2, q)-Solvable Groups
本文为PSL(2,q)-可解群提供素图分类准则，并首次对T属于大（可能无限）群族PSL(2,2^f)的T-可解群证明一般分类结果。
We say that finite groups are isospectral if they have the same sets of orders of elements. It is known that every nonsolvable finite group $G$ isospectral to a finite simple group has a unique nonabelian composition factor, that is, the quotient of $G$ by the solvable radical of $G$ is an almost simple group. The main goal of this paper is prove that this almost simple group is a cyclic extension of its socle. To this end, we consider a general situation when $G$ is an arbitrary group with unique nonabelian composition factor, not necessarily isospectral to a simple group, and study the prime…
We study equations in groups G with unique m-th roots for each positive integer m. A word equation in two letters is an expression of the form w(X,A) = B, where w is a finite word in the alphabet {X,A}. We think of A,B in G as fixed coefficients, and X in G as the unknown. Certain word equations, such as XAXAX=B, have solutions in terms of radicals, while others such as XXAX = B do not. We obtain the first known infinite families of word equations not solvable by radicals, and conjecture a complete classification. To a word w we associate a polynomial P_w in Z[x,y] in two commuting variables, …
We define minimal fusion systems in a way that every non-solvable fusion system has a section which is minimal. Minimal fusion systems can also be seen as analogs of Thompson's N-groups. In this paper, we consider a minimal fusion system $\mathcal{F}$ on a finite $p$-group $S$ that has a unique maximal $p$-local subsystem containing $N_{\mathcal{F}}(S)$. For an arbitrary prime $p$, we determine the structure of a certain (explicitly described) $p$-local subsystem of $\mathcal{F}$. If $p=2$, this leads to a complete classification of the fusion system $\mathcal{F}$.
We study analogues of the notions from Lie theory of Levi subgroup and Levi decomposition, in the case of groups G definable in an o-minimal expansion of a real closed field. With suitable definitions, we prove that G has a unique maximal ind-definable semisimple subgroup S, up to conjugacy, and that G = RS where R is the solvable radical of G. We also prove that any semisimple subalgebra of the Lie algebra of G corresponds to a unique ind-definable semisimple subgroup of G.
We introduce a new class of lattices, the modernistic lattices, and their duals, the comodernistic lattices. We show that every modernistic or comodernistic lattice has shellable order complex. We go on to exhibit a large number of examples of (co)modernistic lattices. We show comodernism for two main families of lattices that were not previously known to be shellable: the order congruence lattices of finite posets, and a weighted generalization of the k-equal partition lattices. We also exhibit many examples of (co)modernistic lattices that were already known to be shellable. To start with, t…
For a simply connected solvable Lie group G with a cocompact discrete subgroup Γ, we consider the space of differential forms on the solvmanifold G/Γ with values in certain flat bundle so that this space has a structure of a differential graded algebra(DGA). We construct Sullivan's minimal model of this DGA. This result is an extension of Nomizu's theorem for ordinary coefficients in the nilpotent case. By using this result, we refine Hasegawa's result of formality of nilmanifolds and Benson-Gordon's result of hard Lefschetz properties of nilmanifolds.
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