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; among them, the minimal non-solvable groups. We also provide several examples of groups not satisfying the property.
We investigate the palindromic width of finitely generated solvable groups. We prove that every finitely generated $3$-step solvable group has finite palindromic width. More generally, we show the finiteness of palindromic width for finitely generated abelian-by-nilpotent-by-nilpotent groups. For arbitrary solvable groups of step $\geq 3$, we prove that if $G$ is a finitely generated solvable group that is an extension of an abelian group by a group satisfying the maximal condition for normal subgroups, then the palindromic width of $G$ is finite. We also prove that the palindromic width of $\…
We describe a new approach towards the systematic construction of finite groups up to isomorphism. This approach yields a practical algorithm for the construction of finite solvable groups up to isomorphism. We report on a GAP implementation of this method for finite solvable groups and exhibit some sample applications.
In this paper we describe all the finite almost simple groups whose Gruenberg--Kegel graphs coincide with Gruenberg--Kegel graphs of finite solvable groups.
Suppose that a finite solvable group $G$ acts faithfully, irreducibly and quasi-primitively on a finite vector space $V$, and $G$ is not metacyclic. Then $G$ always has a regular orbit on $V$ except for a few "small" cases. We completely classify these cases in this paper.
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.
Let $G$ be a finite non-solvable group with solvable radical $Sol(G)$. The solvable graph $Γ_s(G)$ of $G$ is a graph with vertex set $G\setminus Sol(G)$ and two distinct vertices $u$ and $v$ are adjacent if and only if $\langle u, v \rangle$ is solvable. We show that $Γ_s (G)$ is not a star graph, a tree, an $n$-partite graph for any positive integer $n \geq 2$ and not a regular graph for any non-solvable finite group $G$. We compute the girth of $Γ_s (G)$ and derive a lower bound of the clique number of $Γ_s (G)$. We prove the non-existence of finite non-solvable groups whose solvable graphs …
We prove a general dichotomy theorem for groups of finite Morley rank with solvable local subgroups and of Prüfer p-rank at least 2, leading either to some p-strong embedding, or to the Prüfer p-rank being exactly 2.
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 $3$.
We provide polynomial lower bounds for residual finiteness of residually finite, finitely generated solvable groups that admit infinite order elements in the Fitting subgroup of strict distortion at least exponential. For this class of solvable groups which include polycyclic groups with a nontrivial exponential radical and the metabelian Baumslag-Solitar groups, we improve the lower bounds found in the literature. Additionally, for the class of residually finite, finitely generated solvable groups of infinite Prüfer rank that satisfy the conditions of our theorem, we provide the first nontriv…
The following result is received: Let $H$ be a non-normal maximal subgroup of a finite solvable group $G$ and let $q \in π(F(H/\mathrm{Core}_GH))$, then $G$ has a Sylow $q$-subgroup $Q$ such that $N_{G}(Q) \subseteq H$.
We study the asymptotic behaviour of the orbit-counting function and a dynamical Mertens' theorem for the full $G$-shift for a finitely-generated torsion-free nilpotent group $G$. Using bounds for the M{ö}bius function on the lattice of subgroups of finite index and known subgroup growth estimates, we find a single asymptotic of the shape \[ \sum_{|τ|\le N}\frac{1}{e^{h|τ|}}\sim CN^α(\log N)^β \] where $|τ|$ is the cardinality of the finite orbit $τ$. For the usual orbit-counting function we find upper and lower bounds together with numerical evidence to suggest that for actions of non-cyclic …
We prove that, for a finitely generated residually finite group, having solvable word problem is not a sufficient condition to be a subgroup of a finitely presented residually finite group. The obstruction is given by a residually finite group with solvable word problem for which there is no effective method that allows, given some non-identity element, to find a morphism onto a finite group in which this element has a non-trivial image. We also prove that the depth function of this group grows faster than any recursive function.
Let $G$ be a finite group and let $π$ be a set of primes. In this paper, we prove a criterion for the existence of a solvable $π$-Hall subgroup of $G$, precisely, the group $G$ has a solvable $π$-Hall subgroup if, and only if, $G$ has a $\{p,q\}$-Hall subgroup for any pair $p$, $q\inπ$.
We prove that the p-Quillen complex of a finite solvable group with cyclic derived group is Cohen-Macaulay, if p is an odd prime. If p = 2 we prove a similar conclusion, but there is a discussion to be made.
We lay down the fundations of the theory of groups of finite Morley rank in which local subgroups are solvable and we proceed to the local analysis of these groups. We prove the main Uniqueness Theorem, analogous to the Bender method in finite group theory, and derive its corollaries. We also consider homogeneous cases as well as torsion.
We prove that if a solvable group A acts coprimely on a solvable group G, then A has a "large" orbit in its corresponding action on the set of ordinary complex irreducible characters of G. This extends (at the cost of a weaker bound) a 2005 result of A. Moreto who obtained such a bound in case that A is a p-group.
In this paper we show that if S is a simple classical group, a group G is contained in inner-diagonal automorphisms of S and contains S, and H is a solvable Hall subgroup of G, then there exists five conjugates of H, whose intersection is trivial.
Let $G$ be a finite group. For $x \in G$, we define the solvabilizer of $x$ in $G$, denoted $sol_G(x)$, to be the set $\{g \in G \mid \langle g,x \rangle$ is solvable$\}$. A group $G$ is an S-group if $sol_G(x)$ is a subgroup of $G$ for every $x \in G$. In this paper we prove that $G$ is solvable $\Leftrightarrow$ $G$ is an S-group. Secondly, we define the non-solvable graph of $G$ (denoted ${\mathcal S}_{G}$). Its vertices are $G$ and there is an edge between $x,y \in G$ whenever $\langle x,y \rangle$ is not solvable. If $S(G)$ is the solvable radical of $G$ and $G$ is not solvable, we look a…
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 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…
We compute the Bieri-Neumann-Strebel invariants $Σ^1$ for the generalized solvable Baumslag-Solitar groups $Γ_n$ and their finite index subgroups. Using $Σ^1$, we show that certain finite index subgroups of $Γ_n$ cannot be isomorphic to $Γ_{k}$ for any $k$. In addition, we use the BNS-invariants to give a new proof of property $R_\infty$ for the groups $Γ_n$ and their finite index subgroups.
In this paper, we investigate the M{ö}bius function $μ\_{\mathcal{S}}$ associated to a (locally finite) poset arising from a semigroup $\mathcal{S}$ of $\mathbb{Z}^m$. We introduce and develop a new approach to study $μ\_{\mathcal{S}}$ by using the Hilbert series of $\mathcal{S}$. The latter enables us to provide formulas for $μ\_{\mathcal{S}}$ when $\mathcal{S}$ belongs to certain families of semigroups. Finally, a characterization for a locally finite poset to be isomorphic to a semigroup poset is given.
We refer to the set of the orders of elements of a finite group as its spectrum and say that groups are isospectral if their spectra coincide. We prove that with the only specific exception the solvable radical of a nonsolvable finite group isospectral to a finite simple group is nilpotent.
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 prove the following version of Milnor's theorem on solvable groups of exponential growth: A finitely generated solvable group which is not polycyclic contains an ascending HNN extension. Consequently, a finitely generated solvable group is ERF iff it is polycyclic.
Suppose that a finite solvable group $G$ acts faithfully, irreducibly and quasi-primitively on a finite vector space $V$. Then $G$ has a uniquely determined normal subgroup $E$ which is a direct product of extraspecial $p$-groups for various $p$ and we denote $e=\sqrt{|E/\bZ(E)|}$. We prove that when $e=2,3,4,8,9,16$, $G$ will have regular orbits on $V$ when the corresponding vector space is not too small.
A primary covering of a finite group $G$ is a family of proper subgroups of $G$ whose union contains the set of elements of $G$ having order a prime power. We denote with $σ_0(G)$ the smallest size of a primary covering of $G$, and call it the primary covering number of $G$. We study this number and compare it with its analogous $σ(G)$, the covering number, for the classes of groups $G$ that are solvable and symmetric.
We are looking for the smallest integer k>1 providing the following characterization of the solvable radical R(G) of any finite group G: R(G) coincides with the collection of all g such that for any k elements a_1,a_2,...,a_k the subgroup generated by the elements g, a_iga_i^{-1}, i=1,...,k, is solvable. We consider a similar problem of finding the smallest integer l>1 with the property that R(G) coincides with the collection of all g such that for any l elements b_1,b_2,...,b_l the subgroup generated by the commutators [g,b_i], i=1,...,l, is solvable. Conjecturally, k=l=3. We prove that both …
Let $G$ be a finite solvable group, given through a refined consistent polycyclic presentation, and $α$ an automorphism of $G$, given through its images of the generators of $G$. In this paper, we discuss algorithms for computing the order of $α$ as well as the cycle length of a given element of $G$ under $α$. We give correctness proofs and discuss the theoretical complexity of these algorithms. Along the way, we carry out detailed complexity analyses of several classical algorithms on finite polycyclic groups.
We prove that an element $g$ of prime order $>3$ belongs to the solvable radical $R(G)$ of a finite (or, more generally, a linear) group if and only if for every $x\in G$ the subgroup generated by $g, xgx^{-1}$ is solvable. This theorem implies that a finite (or a linear) group $G$ is solvable if and only if in each conjugacy class of $G$ every two elements generate a solvable subgroup.
In this paper, we prove a series of results on group embeddings in groups with a small number of generators. We show that each finitely generated group $G$ lying in a variety ${\mathcal M}$ can be embedded in a $4$-generated group $H \in {\mathcal M}{\mathcal A}$ (${\mathcal A}$ means the variety of abelian groups). If $G$ is a finite group, then $H$ can also be found as a finite group. It follows, that any finitely generated (finite) solvable group $G$ of the derived length $l$ can be embedded in a $4$-generated (finite) solvable group $H$ of length $l+1$. Thus, we answer the question of V. H…
A group has finite palindromic width if there exists $n$ such that every element can be expressed as a product of $n$ or fewer palindromic words. We show that if $G$ has finite palindromic width with respect to some generating set, then so does $G \wr \mathbb{Z}^{r}$. We also give a new, self-contained, proof that finitely generated metabelian groups have finite palindromic width. Finally, we show that solvable groups satisfying the maximal condition on normal subgroups (max-n) have finite palindromic width.
We prove that every countable group with solvable power problem embeds into a finitely presented 2-generated group with solvable power and conjugacy problems.
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…
Let $F$ be either a free nilpotent group of a given class and of finite rank or a free solvable group of a certain derived length and of finite rank. We show precisely which ones have the $R_{\infty}$ property. Finally, we also show that the free group of infinite rank does not have the $R_{\infty}$ property.
Let $p$ be a prime such that $p \geq 5$. Let $G$ be a finite $p$-solvable group and let $p^a$ be the largest power of $p$ dividing $χ(1)$ for an irreducible character $χ$ of $G$, we show that $|G:F(G)|_p \leq p^{5.5a}$. Let $G$ be a finite $p$-solvable group with trivial maximal normal solvable subgroup and we denote $|G|_p=p^n$, then $G$ contains a block of defect less than or equal to $\lfloor \frac {2n} {3} \rfloor$.
We prove that the solvable radical of a finite group G coincides with the set of elements y having the following property: for any x in G the subgroup of G generated by x and y is solvable. We present analogues of this result for finite dimensional Lie algebras and some classes of infinite groups. We also consider a similar problem for pairs of elements.
A classical theorem on character degrees states that if a finite group has fewer than four character degrees, then the group is solvable. We prove a corresponding result on character values by showing that if a finite group has fewer than eight character values in its character table, then the group is solvable. This confirms a conjecture of T. Sakurai. We also classify non-solvable groups with exactly eight character values.
The non-solvable graph of a finite group G is a simple graph whose vertices are the elements of G and there is an edge between x and y if and only if the subgroup generated by x and y is not solvable. The isolated vertices in the non-solvable graph are exactly the elements of the solvable radical of G. Given a finite group G and element x of G, the solvabilizer of x with respect to G is the set of all elements y of G such that the subgroup generated by x and y is solvable. The purpose of this paper is to study some properties of the non-solvable graph and the structure of the solvabilizer of x…
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.
