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 k and l are at most 7. In particular, this means that a finite group G is solvable if and only if in each conjugacy class of G every 8 elements generate a solvable subgroup.
A gyrogroup is a nonassociative group-like structure modelled on the space of relativistically admissible velocities with a binary operation given by Einstein's velocity addition law. In this article, we present a few of groups sitting inside a gyrogroup $G$, including the commutator subgyrogroup, the left nucleus, and the radical of $G$. The normal closure of the commutator subgyrogroup, the left nucleus, and the radical of $G$ are in particular normal subgroups of $G$. We then give a criterion to determine when a subgyrogroup $H$ of a finite gyrogroup $G$, where the index $[G\colon H]$ is th…
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.
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.
Guralnick, Kunyavskii, Plotkin and Shalev have shown that the solvable radical of a finite group $G$ can be characterized as the set of all $x\in G$ such that $<x,y>$ is solvable for all $y\in G$. We prove two generalizations of this result. Firstly, it is enough to check the solvability of $<x,y>$ for every $p$-element $y\in G$ for every odd prime $p$. Secondly, if $x$ has odd order, then it is enough to check the solvability of $<x,y>$ for every 2-element $y\in G$.
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 prove that there exists a constant $k$ with the property: if $\calC$ is a conjugacy class of a finite group $G$ such that every $k$ elements of $\calC$\ generate a solvable subgroup then $\calC$ generates a solvable subgroup. In particular, using the Classification of Finite Simple Groups, we show that we can take $k=4$. We also present proofs that do not use the Classification theorem. The most direct proof gives a value of $k=10$. By lengthening one of our arguments slightly, we obtain a value of $k=7$.
The study of associativity of the commutator operation in groups goes back to some work of Levi in 1942. In the 1960's Richard J. Thompson created a group F whose elements are representatives of the generalized associative law for an arbitrary binary operation. In 2006, Geoghegan and Guzman proved that a group G is solvable if and only if the commutator operation in G eventually satisfies ALL instances of the associative law, and also showed that many non-solvable groups do not satisfy any instance of the generalized associative law. We will address the question: Is there a non-solvable group …
We prove the following: For any $ρ\in\left(0,1\right)$, if a finite group $G$ has an automorphism with a cycle of length at least $ρ\cdot|G|$, then the index of the solvable radical $\operatorname{Rad}(G)$ in $G$ is bounded from above in terms of $ρ$, and such a condition is strong enough to imply solvability of $G$ if and only if $ρ>\frac{1}{10}$. Furthermore, considering, for exponents $e\in\left(0,1\right)$, the condition that a finite group $G$ have an automorphism with a cycle of length at least $|G|^e$, such a condition is strong enough to imply $|\operatorname{Rad}(G)|\to\infty$ for $|G…
Every commutator preserving bijection of the unipotent radical $Up(2n, R)$ of the Borel subgroup of the classical symplectic group of rank at least 4 over a field $F$ such that $6F=F$ is shown to be the composition of a standard automorphism of $Up(2n,R)$ and a central map. The latter is a bijection which acts as the right multiplication by a matrix from the center of $Up(2n,F)$.
Let $G$ be a connected exponential Lie group and $R$ be the solvable radical of $G$. We describe a condition on $G/R$ under which one can then conclude that $R$ is an exponential Lie group. The condition holds in particular when $G$ is a complex Lie group and this yields a stronger version of a result of Moskowitz and Sacksteder \cite{MS} on the center of a complex exponential Lie group being connected. Along the way we prove a criterion for elements from certain subsets of a solvable Lie group to be exponential, which would be of independent interest.
This work provides a method(an algorithm) for solving the solvable unary algebraic equation $f(x)=0$ ($f(x)\in\mathbb{Q}[x]$) of arbitrary degree and obtaining the exact radical roots. This method requires that we know the Galois group as the permutation group of the roots of $f(x)$ and the approximate roots with sufficient precision beforehand. Of course, the approximate roots are not necessary but can help reduce the quantity of computation. The algorithm complexity is approximately proportional to the 4th power of the size of the Galois group of $f(x)$. The whole algorithm doesn't need to d…
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, …
This note aims to clarify what conditions on a connected Lie group G imply that its maximal connected normal solvable subgroup R intersects each lattice of G as a lattice in R.
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 $G$.
Suppose that G is a finite group and x in G has prime order p > 3. Then x is contained in the solvable radical of G if (and only if) <x,x^g> is solvable for all g in G. If G is an almost simple group and x in G has prime order p > 3 then this implies that there exists g in G such that <x,x^g> is not solvable. In fact, this is also true when p=3 with very few exceptions, which are described explicitly.
A classical theorem of R. Baer describes the nilpotent radical of a finite group G as the set of all Engel elements, i.e. elements y in G such that for any x in G the n-th commutator [x,y,...,y] equals 1 for n big enough. We obtain a characterization of the solvable radical of a finite dimensional Lie algebra defined over a field of characteristic zero in similar terms. We suggest a conjectural description of the solvable radical of a finite group as the set of Engel-like elements and reduce this conjecture to the case of a finite simple group.
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.
In this paper we deal with the class C of decomposable solvable Lie groups having dimension at most six. We determine those Lie groups in C and their subgroups which are the multiplication group Mult(L) and the inner mapping group Inn(L) for three-dimensional connected simply connected topological loops L. These loops L have one- or two-dimensional centre and their group Mult(L) has two- or three-dimensional commutator subgroup. Together with this result we obtain that every at most 3-dimensional connected topological proper loop having a solvable Lie group of dimension at most six as its mult…
Based on the recent development of commutator theory for loops, we provide both syntactic and semantic characterization of abelian normal subloops. We highlight the analogies between well known central extensions and central nilpotence on one hand, and abelian extensions and congruence solvability on the other hand. In particular, we show that a loop is congruence solvable (that is, an iterated abelian extension of commutative groups) if and only if it is not Boolean complete, reaffirming the connection between computational complexity and solvability. Finally, we briefly discuss relations bet…
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 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…
We observe a correspondence between collections of closed subgroups and normal subgroups in totally disconnected locally compact groups. This correspondence is applied to prove structure theorems for two classes of totally disconnected locally compact second countable groups: the class of such groups with an open solvable subgroup and the class of such groups with a pro-nilpotent compact open subgroup. As a second application, we give new proofs and generalizations of results of G. Willis and Y. Barnea, M. Ershov, and T. Weigel on totally disconnected locally compact groups which are topologic…
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 …
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…
We classify measures on a homogeneous space which are invariant under a certain solvable subgroup and ergodic under its unipotent radical. Our treatment is independent of characteristic. As a result we get the first measure classification for the action of semisimple subgroups without any characteristic restriction.
There are various results in the literature which are part of the general philosophy that a finite group for which a certain parameter (for example, the number of conjugacy classes or the maximum number of elements inverted, squared or cubed by a single automorphism) is large enough must be close to being abelian. In this paper, we show the following: Fix a real number $ρ$ with $0<ρ\leq 1$. Then a finite group $G$ with an automorphism inverting or squaring at least $ρ|G|$ of the elements in $G$ is "almost abelian" in the sense that both the index and the derived length of the solvable radical …
Groups that can be approximated by finite groups have been the center of much research. This has led to the investigations of the subgroups of metric ultraproducts of finite groups. This paper attempts to study the dual problem: what are the quotients of ultraproducts of finite groups? Since an ultraproduct is an abstract quotient of the direct product, this also led to a more general question: what are the abstract quotients of profinite groups? Certain cases were already well-studied. For example, if we have a pro-solvable group, then it is well-known that any finite abstract quotients are s…
Engel groups and Engel elements became popular in 50s. We consider in the paper the more general nil-groups and nil-elements in groups. All these notions are related to nilpotent groups and nilpotent radicals in groups. These notions generate problems which are parallel to Burnside problems for periodic groups. The first three theorems of the paper are devoted to nil-groups and Engel groups, while the other results are connected with the further generalizations. These generalizations extend the theory to solvable groups and solvable radicals in groups. The paper has two parts. The first one (s…
We investigate the relationship between various isomorphism invariants for finite groups. Specifically, we use the Weisfeiler-Leman dimension (WL) to characterize, compare and quantify the effectiveness and complexity of invariants for group isomorphism. It turns out that a surprising number of invariants and characteristic subgroups that are classic to group theory can be detected and identified by a low dimensional Weisfeiler-Leman algorithm. These include the center, the inner automorphism group, the commutator subgroup and the derived series, the abelian radical, the solvable radical, the …
Let $G$ be a group containing a nilpotent normal subgroup $N$ with central series $\{N_j\}$, such that each $N_j/N_{j+1}$ is a $\mathbb{F}$-vector space over a field $\mathbb{F}$ and the action of $G$ on $N_j/N_{j+1}$ induced by the conjugation action is $\mathbb{F}$-linear. For $k\in \mathbb N$ we describe a necessary and sufficient condition for all elements from any coset $xN$, $x\in G$, to admit $k$-th roots in $G$, in terms of the action of $x$ on the quotients $N_j/N_{j+1}.$ This yields in particular a condition for surjectivity of the power maps, generalising various results known in sp…
Let $G$ be a finite group, and assume that $G$ has an automorphism of order at least $ρ|G|$, with $ρ\in\left(0,1\right)$. Generalizing recent analogous results of the author on finite groups with a large automorphism cycle length, we prove that if $ρ>1/2$, then $G$ is abelian, and if $ρ>1/10$, then $G$ is solvable, whereas in general, the assumption implies $[G:\operatorname{Rad}(G)]\leqρ^{-1.78}$, where $\operatorname{Rad}(G)$ denotes the solvable radical of $G$. Furthermore, we generalize an example of Horoševski\uı to show that in finite groups, the quotient of the maximum automorphism orde…
Every finite non-nilpotent group can be extended by a term operation such that solving equations in the resulting algebra is NP-complete and checking identities is co-NP-complete. This result was firstly proven by Horváth and Szabó; the term constructed in their proof depends on the underlying group. In this paper we provide a uniform term extension that induces hardness. In doing so we also characterize a big class of solvable, non-nilpotent groups for which extending by the commutator operation suffices.
We prove that for every $ε> 0$ there exists a $δ> 0$ so that every group of order $n \geq 3$ has at least $δ\log_{2} n/{(\log_{2} \log_{2} n)}^{3+ε}$ conjugacy classes. This sharpens earlier results of Pyber and Keller. Bertram speculates whether it is true that every finite group of order $n$ has more than $\log_{3}n$ conjugacy classes. We answer Bertram's question in the affirmative for groups with a trivial solvable radical.
If an outer (multilinear) commutator identity holds in a large subgroup of a group, then it holds also in a large characteristic subgroup. Similar assertions are valid for algebras and their ideals or subspaces. Varying the meaning of the word "large", we obtain many interesting facts. These results cannot be extended to arbitrary (non-multilinear) identities. As an application, we give a sharp estimate for the `virtual derived length' of (virtually solvable)-by-(virtually solvable) groups.
We prove that the order of a finite group $G$ with trivial solvable radical is bounded above in terms of ${\rm acd}(G)$, the average degree of the irreducible characters. It is not true that the index of the Fitting subgroup is bounded above in terms of ${\rm acd}(G)$, but we show that in certain cases it is bounded in terms of the degrees of the irreducible characters of $G$ that lie over a linear character of the Fitting subgroup. This leads us to propose a refined version of Gluck's conjecture.
It is a long-standing open problem raised by Starostin to describe all finite groups with soluble centralizers of involutions. One can observe that if the centralizer fusion system of an involution is nilpotent, then the centralizer of that involution is soluble. In this paper, we classify the cases when the centralizer fusion system of a central involution in a finite group whose all involutions have soluble centralizers is a nilpotent fusion system. Indeed, we analyse the case when the solvable radical has odd order and the corresponding factor group is simple.
The isomorphism problem for finite groups of order n (GpI) has long been known to be solvable in $n^{\log n+O(1)}$ time, but only recently were polynomial-time algorithms designed for several interesting group classes. Inspired by recent progress, we revisit the strategy for GpI via the extension theory of groups. The extension theory describes how a normal subgroup N is related to G/N via G, and this naturally leads to a divide-and-conquer strategy that splits GpI into two subproblems: one regarding group actions on other groups, and one regarding group cohomology. When the normal subgroup N …
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 study properties and the structure of Cartan subgroups in a connected Lie group. We obtain a characterisation of Cartan subgroups which generalises Wüstner's structure theorem for the same. We show that Cartan subgroups are same as those of the centralizers of maximal compact subgroups of the radical. Moreover, we describe a recipe for constructing Cartan subgroups containing certain nilpotent subgroups in a connected solvable Lie group. We characterise the Cartan subgroups in the quotient group modulo a closed normal subgroup as the images of the Cartan subgroups in the ambient group. We a…
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…
