Engel-type subgroups and length parameters of finite groups

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],..., g] over x ∈ 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 generalized Fitting height. The generalized Fitting height h*(H) of a finite group H is the least number h such that F_h* (H) = H, where F₀* (H) = 1, and F_i+1(H)* is the inverse image of the generalized Fitting subgroup F*(H/F*_i (H)). Let m be the number of prime factors of |g| counting multiplicities. It is proved that if, for some n, the generalized Fitting height of E _n (g) is equal to k, then g belongs to F*_f(k,m)(G), where f(k, m) depends only on k and m. The nonsoluble length λ(H) of a finite group H is defined as the minimum number of nonsoluble factors in a normal series each of whose factors either is soluble or is a direct product of nonabelian simple groups. It is proved that if λ(E _n (g)) = k, then g belongs to a normal subgroup whose nonsoluble length is bounded in terms of k and m. We also state conjectures of stronger results independent of m and show that these conjectures reduce to a certain question about automorphisms of direct products of finite simple groups.     

On the length of a finite group and of its 2-generator subgroups

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) ≤ k for every 2-generator subgroup J of G, then λ(G) ≤ k. It is conjectured that if h* (J) ≤ k for every 2-generator subgroup J, then h* (G) ≤ k. We prove that if h* (〈x, x^g 〉) ≤ k for allx, g ∈ G such that 〈x, x^g 〉 is soluble, then h* (G) is k-bounded.     

Conditional connectivity of bubble sort graphs

A subset F ? V (G) is called an R ^k -vertex-cut of a graph G if G ? F is disconnected and each vertex of G ? F has at least k neighbors in G ? F. The R ^k -vertex-connectivity of G, denoted by κ ^k (G), is the cardinality of a minimum R ^k -vertex-cut of G. Let B _n be the bubble sort graph of dimension n. It is known that κ _k (B _n ) = 2 ^k (n ? k ? 1) for n ≥ 2k and k = 1, 2. In this paper, we prove it for k = 3 and conjecture that it is true for all k ∈ N. We also prove that the connectivity cannot be more than conjectured.     

Distance domination of generalized de Bruijn and Kautz digraphs

Let G = (V,A) be a digraph and k ≥ 1 an integer. For u, v ∈ V, we say that the vertex u distance k-dominate v if the distance from u to v at most k. A set D of vertices in G is a distance k-dominating set if each vertex of V D is distance k-dominated by some vertex of D. The distance k-domination number of G, denoted by γ _k (G), is the minimum cardinality of a distance k-dominating set of G. Generalized de Bruijn digraphs G _B (n, d) and generalized Kautz digraphs G _K (n, d) are good candidates for interconnection networks. Denote Δ _k := (∑ _j=0 ^k d ^j )^?1. F. Tian and J. Xu showed that ?nΔ _k ? γ _k (G _B (n, d)) ≤?n/d ^k? and ?nΔ _k ? ≤ γ _k (G _K (n, d)) ≤ ?n/d ^k ?. In this paper, we prove that every generalized de Bruijn digraph G _B (n, d) has the distance k-domination number ?nΔ _k ? or ?nΔ _k ?+1, and the distance k-domination number of every generalized Kautz digraph G _K (n, d) bounded above by ?n/(d ^k?1+d ^k )?. Additionally, we present various sufficient conditions for γ _k (G _B (n, d)) = ?nΔ _k ? and γ _k (G _K (n, d)) = ?nΔ _k ?.     

Ideal Spaces of Measurable Operators Affiliated to A Semifinite Von Neumann Algebra

Suppose that M is a von Neumann algebra of operators on a Hilbert space H and τ is a faithful normal semifinite trace on M. Let E, F and G be ideal spaces on (M, τ). We find when a τ-measurable operator X belongs to E in terms of the idempotent P of M. The sets E+F and E·F are also ideal spaces on (M, τ); moreover, E·F = F·E and (E+F)·G = E·G+F·G. The structure of ideal spaces is modular. We establish some new properties of the L₁(M, τ) space of integrable operators affiliated to the algebra M. The results are new even for the *-algebra M = B(H) of all bounded linear operators on H which is endowed with the canonical trace τ = tr.     

Normal coverings of solvable groups

For a finite non cyclic group G, let γ(G) be the smallest integer k such that G contains k proper subgroups H ₁, . . . , H _k with the property that every element of G is contained in \({H_i^g}\) for some \({i \in \{1,\dots,k\}}\) and \({g \in G.}\) We prove that for every n ≥ 2, there exists a finite solvable group G with γ(G) = n.     

On a classical theorem on the diameter and minimum degree of a graph

In this work, we obtain good upper bounds for the diameter of any graph in terms of its minimum degree and its order, improving a classical theorem due to Erd¨os, Pach, Pollack and Tuza.We use these bounds in order to study hyperbolic graphs(in the Gromov sense). To compute the hyperbolicity constant is an almost intractable problem, thus it is natural to try to bound it in terms of some parameters of the graph. Let H(n, δ_0) be the set of graphs G with n vertices and minimum degree δ_0, and J(n, Δ) be the set of graphs G with n vertices and maximum degree Δ. We study the four following extremal problems on graphs: a(n, δ_0) = min{δ(G) | G ∈ H(n, δ_0)}, b(n, δ_0) = max{δ(G) |G ∈ H(n, δ_0)}, α(n, Δ) = min{δ(G) | G ∈ J(n, Δ)} and β(n, Δ) = max{δ(G) | G ∈ J(n, Δ)}. In particular, we obtain bounds for b(n, δ_0) and we compute the precise value of a(n, δ_0), α(n, Δ) andβ(n, Δ) for all values of n, δ_0 and Δ, respectively.     

Supersolubility of a Finite Group with Normally Embedded Maximal Subgroups in Sylow Subgroups

Let P be a subgroup of a Sylow subgroup of a finite group G. If P is a Sylow subgroup of some normal subgroup of G then P is called normally embedded in G. We establish tests for a finite group G to be p-supersoluble provided that every maximal subgroup of a Sylow p-subgroup of X is normally embedded in G. We study the cases when X is a normal subgroup of G, X = O_p',p(H), and X = F*(H) where H is a normal subgroup of G.     

Removable Singularities of Solutions of the Minimal Surface Equation

Suppose that G is a bounded domain in ? ⁿ (n ? 2), E ≠ G is a relatively closed set in G, and 0 < α < 1. We prove that E is removable for solutions of the minimal surface equation in the class C ^1,α(G)_loc if and only if the (n ? 1 + α)-dimensional Hausdorff measure of E is zero.     

Galois theory and Lubin-Tate cochains on classifying spaces

We consider brave new cochain extensions F(BG ₊,R) → F(EG ₊,R), where R is either a Lubin-Tate spectrum E _n or the related 2-periodic Morava K-theory K _n , and G is a finite group. When R is an Eilenberg-Mac Lane spectrum, in some good cases such an extension is a G-Galois extension in the sense of John Rognes, but not always faithful. We prove that for E _n and K _n these extensions are always faithful in the K _n local category. However, for a cyclic p-group \(C_{p^r } \), the cochain extension \(F(BC_{p^r + } ,E_n ) \to F(EC_{p^r + } ,E_n )\) is not a Galois extension because it ramifies. As a consequence, it follows that the E _n -theory Eilenberg-Moore spectral sequence for G and BGdoes not always converge to its expected target.     

Extendability of Contractible Configurations for Nowhere-Zero Flows and Modulo Orientations

Let H be a connected graph and G be a supergraph of H. It is trivial that for any k-flow (D, f) of G, the restriction of (D, f) on the edge subset E(G / H) is a k-flow of the contracted graph G / H. However, the other direction of the question is neither trivial nor straightforward at all: for any k-flow \((D',f')\) of the contracted graph G / H, whether or not the supergraph G admits a k-flow (D, f) that is consistent with \((D',f')\) in the edge subset E(G / H). In this paper, we will investigate contractible configurations and their extendability for integer flows, group flows, and modulo orientations. We show that no integer flow contractible graphs are extension consistent while some group flow contractible graphs are also extension consistent. We also show that every modulo \((2k+1)\)-orientation contractible configuration is also extension consistent and there are no modulo (2k)-orientation contractible graphs.     

Degree sum of a pair of independent edges and <Emphasis Type="Italic">Z</Emphasis><Subscript>3</Subscript>-connectivity

Let G be a 2-edge-connected simple graph on n vertices. For an edge e = uv ∈ E(G), define d(e) = d(u) + d(v). Let F denote the set of all simple 2-edge-connected graphs on n ≥ 4 vertices such that G ∈ F if and only if d(e) + d(e’) ≥ 2n for every pair of independent edges e, e’ of G. We prove in this paper that for each G ∈ F, G is not Z ₃-connected if and only if G is one of K _2,n?2, K _3,n?3, K _2,n?2 ⁺ , K _3,n?3 ⁺ or one of the 16 specified graphs, which generalizes the results of X. Zhang et al. [Discrete Math., 2010, 310: 3390–3397] and G. Fan and X. Zhou [Discrete Math., 2008, 308: 6233–6240].     

Semistability of Frobenius Direct Image of Representations of Cotangent Bundles

Let k be an algebraically closed field of characteristic p > 0, X a smooth projective variety over k with a fixed ample divisor H, F_X : X → X the absolute Frobenius morphism on X. Let E be a rational GL_n(k)-bundle on X, and ρ : GL_n(k) → GL_m(k) a rational GL_n(k)-representation of degree at most d such that ρ maps the radical RGL_n(k)) of GL_n(k) into the radical R(GL_m(k)) of GL_m(k). We show that if \(F_X^{N*}(E)\) is semistable for some integer \(N \ge {\max {_{0 < r < m}}}(_r^m) \cdot {\log _p}(dr)\), then the induced rational GL_m(k)-bundle E(GL_m(k)) is semistable. As an application, if dimX = n, we get a sufficient condition for the semistability of Frobenius direct image \(F_{X*}(\rho*(\Omega_X^1))\), where \(\rho*(\Omega_X^1)\) is the vector bundle obtained from \(\Omega_X^1\) via the rational representation ρ.     

Neighborhood Conditions for Fractional ID-<Emphasis Type="Italic">k</Emphasis>-factor-critical Graphs

Let G be a graph and k ≥ 2 a positive integer. Let h: E(G) → [0, 1] be a function. If \(\sum\limits_{e \mathrel\backepsilon x} {h(e) = k} \) holds for each x ∈ V (G), then we call G[F_h] a fractional k-factor of G with indicator function h where F_h = {e ∈ E(G): h(e) > 0}. A graph G is fractional independent-set-deletable k-factor-critical (in short, fractional ID-k-factor-critical), if G ? I has a fractional k-factor for every independent set I of G. In this paper, we prove that if n ≥ 9k ? 14 and for any subset X ? V (G) we have

$${N_G}(X) = V(G)if|X| \geqslant \left\lfloor {\frac{{kn}}{{3k - 1}}} \right\rfloor ;or|{N_G}(X)| \geqslant \frac{{3k - 1}}{k}|X|if|X| < \left\lfloor {\frac{{kn}}{{3k - 1}}} \right\rfloor ,$$

then G is fractional ID-k-factor-critical.

     

Generalized 3-edge-connectivity of Cartesian product graphs

The generalized k-connectivity κ _k (G) of a graph G was introduced by Chartrand et al. in 1984. As a natural counterpart of this concept, Li et al. in 2011 introduced the concept of generalized k-edge-connectivity which is defined as λ _k (G) = min{λ(S): S ? V (G) and |S| = k}, where λ(S) denotes the maximum number l of pairwise edge-disjoint trees T ₁, T ₂, …, T _l in G such that S ? V (T _i ) for 1 ? i ? l. In this paper we prove that for any two connected graphs G and H we have λ ₃(G □ H) ? λ ₃(G) + λ ₃(H), where G □ H is the Cartesian product of G and H. Moreover, the bound is sharp. We also obtain the precise values for the generalized 3-edge-connectivity of the Cartesian product of some special graph classes.     

Token Graphs

For a graph G and integer k ≥ 1, we define the token graph F _k (G) to be the graph with vertex set all k-subsets of V(G), where two vertices are adjacent in F _k (G) whenever their symmetric difference is a pair of adjacent vertices in G. Thus vertices of F _k (G) correspond to configurations of k indistinguishable tokens placed at distinct vertices of G, where two configurations are adjacent whenever one configuration can be reached from the other by moving one token along an edge from its current position to an unoccupied vertex. This paper introduces token graphs and studies some of their properties including: connectivity, diameter, cliques, chromatic number, Hamiltonian paths, and Cartesian products of token graphs.     

ON THE COADJOINT ORBITS OF MAXIMAL UNIPOTENT SUBGROUPS OF REDUCTIVE GROUPS

Let G be a simple algebraic group defined over an algebraically closed field of characteristic 0 or a good prime for G. Let U be a maximal unipotent subgroup of G and \( \mathfrak{u} \) its Lie algebra. We prove the separability of orbit maps and the connectedness of centralizers for the coadjoint action of U on (certain quotients of) the dual \( \mathfrak{u} \)* of \( \mathfrak{u} \). This leads to a method to give a parametrization of the coadjoint orbits in terms of so-called minimal representatives which form a disjoint union of quasi-affine varieties. Moreover, we obtain an algorithm to explicitly calculate this parametrization which has been used for G of rank at most 8, except E₈.When G is defined and split over the field of q elements, for q the power of a good prime for G, this algorithmic parametrization is used to calculate the number k(U(q); \( \mathfrak{u} \)*(q)) of coadjoint orbits of U(q) on \( \mathfrak{u} \)*(q). Since k(U(q), \( \mathfrak{u} \)*(q)) coincides with the number k(U(q)) of conjugacy classes in U(q), these calculations can be viewed as an extension of the results obtained in [11]. In each case considered here there is a polynomial h(t) with integer coefficients such that for every such q we have k(U(q)) = h(q). We also explain implications of our results for a parametrization of the irreducible complex characters of U(q).     

Asymptotic Analysis for Random Walks with Nonidentically Distributed Jumps Having Finite Variance

Let ξ₁,ξ₂,... be independent random variables with distributions F₁F₂,... in a triangular array scheme (F _i may depend on some parameter). Assume that Eξ _i = 0, Eξ _i ² < ∞, and put \(S_n = \sum {_{i = 1}^n \;} \xi _i ,\;\overline S _n = \max _{k \leqslant n} S_k\). Assuming further that some regularly varying functions majorize or minorize the “averaged” distribution \(F = \frac{1}{n}\sum {_{i = 1}^n F_i }\), we find upper and lower bounds for the probabilities P(S _n > x) and \(P(\bar S_n > x)\). We also study the asymptotics of these probabilities and of the probabilities that a trajectory {S _k } crosses the remote boundary {g(k)}; that is, the asymptotics of P(max_k≤n(S _k ? g(k)) > 0). The case n = ∞ is not excluded. We also estimate the distribution of the first crossing time.     

The boundary of the outer space of a free product

Let G be a countable group that splits as a free product of groups of the form G = G ₁ *···* G _k * F _N , where F _N is a finitely generated free group. We identify the closure of the outer space PO(G, {G ₁,..., G _k }) for the axes topology with the space of projective minimal, very small (G, {G ₁,..., G _k })-trees, i.e. trees whose arc stabilizers are either trivial, or cyclic, closed under taking roots, and not conjugate into any of the G _i ’s, and whose tripod stabilizers are trivial. Its topological dimension is equal to 3N + 2k ? 4, and the boundary has dimension 3N + 2k ? 5. We also prove that any very small (G, {G ₁,..., G _k })-tree has at most 2N + 2k?2 orbits of branch points.