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.     

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.     

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.     

Free groups and ends of graphs

Let X be a connected graph. An automorphism of X is said to be parabolic if it leaves no finite subset of vertices in X invariant and fixes precisely one end of X and hyperbolic if it leaves no finite subset of vertices in X invariant and fixes precisely two ends of X. Various questions concerning dynamics of parabolic and hyperbolic automorphisms are discussed.The set of ends which are fixed by some hyperbolic element of a group G acting on X is denoted by ?(G). If G contains a hyperbolic automorphism of X and G fixes no end of X, then G contains a free subgroup F such that ?(F) is dense in ?(G) with respect to the natural topology on the ends of X.As an application we obtain the following: A group which acts transitively on a connected graph and fixes no end has a free subgroup whose directions are dense in the end boundary.     

Polyboxes,Cube Tilings and Rigidity

A non-empty subset A of X=X ₁×???×X _d is a (proper) box if A=A ₁×???×A _d and A _i ?X _i for each i. Suppose that for each pair of boxes A, B and each i, one can only know which of the three states takes place: A _i =B _i , A _i =X _i ?B _i , A _i ?{B _i ,X _i ?B _i }. Let F and G be two systems of disjoint boxes. Can one decide whether ∪F=∪G? In general, the answer is ‘no’, but as is shown in the paper, it is ‘yes’ if both systems consist of pairwise dichotomous boxes. (Boxes A, B are dichotomous if there is i such that A _i =X _i ?B _i .) Several criteria that enable to compare such systems are collected. The paper includes also rigidity results, which say what assumptions have to be imposed on F to ensure that ∪F=∪G implies F=G. As an application, the rigidity conjecture for 2-extremal cube tilings of Lagarias and Shor is verified.     

Algebraic Hamiltonian actions

In this paper we deal with a Hamiltonian action of a reductive algebraic group G on an irreducible normal affine Poisson variety X. We study the quotient morphism \({\mu_{G,X}//G : X//G \rightarrow \mathfrak{g} //G}\) of the moment map \({\mu_{G,X} : X\rightarrow \mathfrak{g}}\) . We prove that for a wide class of Hamiltonian actions (including, for example, actions on generically symplectic varieties) all fibers of the morphism μ _G,X //G have the same dimension. We also study the “Stein factorization” of μ _G,X //G. Namely, let C _G,X denote the spectrum of the integral closure of \({\mu_{G,X}^{*}(\mathbb{K}[\mathfrak{g}]^G)}\) in \({\mathbb{K}(X)^G}\) . We investigate the structure of the \({\mathfrak{g} //G}\) -scheme C _G,X . Our results partially generalize those obtained by F. Knop for the actions on cotangent bundles and symplectic vector spaces.     

Topological entropy of sets of generic points for actions of amenable groups

Let G be a countable discrete infinite amenable group which acts continuously on a compact metric space X and let μ be an ergodic G-invariant Borel probability measure on X. For a fixed tempered F?lner sequence {Fn} in G with limn→+∞|Fn|/log n= ∞, we prove the following result:h_top~B(G_μ, {F_n}) = h_μ(X, G),where G_μ is the set of generic points for μ with respect to {F_n} and h_top~B(G_μ, {F_n}) is the Bowen topological entropy(along {F_n}) on G_μ. This generalizes the classical result of Bowen(1973).     

Classification of finite groups satisfying a minimal condition

If H is a subgroup of a finite group G then we denote the normal closure of H in G by H ^G . We call G a PE-group if every minimal subgroup X of G satisfies N ^G (X) ∩ X ^G = X. The authors classify the finite non-PE-groups whose maximal subgroups of even order are PE-groups.     

Note: A conjecture on partitions of groups

We conjecture that every infinite group G can be partitioned into countably many cells \(G = \bigcup\limits_{n \in \omega } {A_n }\) such that cov(A _n A _n ^?1 ) = |G| for each n ∈ ω Here cov(A) = min{|X|: X} ? G, G = X A}. We confirm this conjecture for each group of regular cardinality and for some groups (in particular, Abelian) of an arbitrary cardinality.     

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.     

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.     

Additive bases in groups

We study the problem of removing an element from an additive basis in a general abelian group. We introduce analogues of the classical functions X, S and E (defined in the case of ?) and obtain bounds on them. Our estimates on the functions S _G and E _G are valid for general abelian groups G while in the case of X _G we show that distinct types of behaviours may occur depending on G.     

Strongly transitive actions on euclidean buildings

We prove a decomposition result for a group G acting strongly transitively on the Tits boundary of a Euclidean building. As an application we provide a local to global result for discrete Euclidean buildings, which generalizes results in the locally compact case by Caprace–Ciobotaru and Burger–Mozes. Let X be a Euclidean building without cone factors. If a group G of automorphisms of X acts strongly transitively on the spherical building at infinity ?X, then the G-stabilizer of every affine apartment in X contains all reflections along thick walls. In particular G acts strongly transitively on X if X is simplicial and thick.     

Equivariant Euler–Poincaré characteristic in sheaf cohomology

Let X be a Hausdorff space equipped with a continuous action of a finite group G and a G-stable family of supports \({\Phi}\). Fix a number field F with ring of integers R. We study the class \({\chi = \sum_j (-1)^j [H^j_\Phi (X, \mathcal{E}) \otimes_R F]}\) in the character group of G over F for any flat G-sheaf \({\mathcal{E}}\) of R-modules over X. Under natural cohomological finiteness conditions we give a formula for \({\chi}\) with respect to the basis given by the irreducible characters of G. We discuss applications of our result concerning the cohomology of arithmetic groups.     

Subgraph Transversal of Graphs

Given graphs F and G, denote by \({\tau_F}(G)\) the cardinality of a smallest subset \(T {\subseteq}V(G)\) that meets every maximal F-free subgraph of G. Erdös, Gallai and Tuza [9] considered the question of bounding \(\tau_{\overline{K_2}}(G)\) by a constant fraction of |G|. In this paper, we will give a complete answer to the following question: for which F, is τ _F (G) bounded by a constant fraction of |G|?In addition, for those graphs F for which \({\tau_F}(G)\) is not bounded by any fraction of |G|, we prove that \(\tau_F(G)\le|G|-\frac{1}{2}\sqrt{|G|}+\frac{1}{2}\), provided F is not K _k or \(\overline{K_k}\).     

On the finite principal bundles

Let G be a connected linear algebraic group defined over \({\mathbb C}\). Fix a finite dimensional faithful G-module V ₀. A holomorphic principal G-bundle E _G over a compact connected Kähler manifold X is called finite if for each subquotient W of the G-module V ₀, the holomorphic vector bundle E _G (W) over X associated to E _G for W is finite. Given a holomorphic principal G-bundle E _G over X, we prove that the following four statements are equivalent: (1) The principal G-bundle E _G admits a flat holomorphic connection whose monodromy group is finite. (2) There is a finite étale Galois covering \({f: Y \longrightarrow X}\) such that the pullback f*E _G is a holomorphically trivializable principal G-bundle over Y. (3) For any finite dimensional complex G-module W, the holomorphic vector bundle E _G (W) = E ×  ^G W over X, associated to the principal G-bundle E _G for the G-module W, is finite. (4) The principal G-bundle E _G is finite.     

Abelianization of the <Emphasis Type="Italic">F</Emphasis>-divided fundamental group scheme

Let (X , x ₀) be a pointed smooth proper variety defined over an algebraically closed field. The Albanese morphism for (X , x ₀) produces a homomorphism from the abelianization of the F-divided fundamental group scheme of X to the F-divided fundamental group of the Albanese variety of X. We prove that this homomorphism is surjective with finite kernel. The kernel is also described.     

On Lipschitz extension from finite subsets

We prove that for every n ∈ ? there exists a metric space (X, d _X), an n-point subset S ? X, a Banach space (Z, \({\left\| \right\|_Z}\)) and a 1-Lipschitz function f: S → Z such that the Lipschitz constant of every function F: X → Z that extends f is at least a constant multiple of \(\sqrt {\log n} \). This improves a bound of Johnson and Lindenstrauss [JL84]. We also obtain the following quantitative counterpart to a classical extension theorem of Minty [Min70]. For every α ∈ (1/2, 1] and n ∈ ? there exists a metric space (X, d _X), an n-point subset S ? X and a function f: S → ?₂ that is α-Hölder with constant 1, yet the α-Hölder constant of any F: X → ?₂ that extends f satisfies \({\left\| F \right\|_{Lip\left( \alpha \right)}} > {\left( {\log n} \right)^{\frac{{2\alpha - 1}}{{4\alpha }}}} + {\left( {\frac{{\log n}}{{\log \log n}}} \right)^{{\alpha ^2} - \frac{1}{2}}}\). We formulate a conjecture whose positive solution would strengthen Ball’s nonlinear Maurey extension theorem [Bal92], serving as a far-reaching nonlinear version of a theorem of König, Retherford and Tomczak-Jaegermann [KRTJ80]. We explain how this conjecture would imply as special cases answers to longstanding open questions of Johnson and Lindenstrauss [JL84] and Kalton [Kal04].     

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.

     

Symplectic <Emphasis Type="Italic">S</Emphasis><Subscript>5</Subscript> action on symplectic homotopy K3 surfaces

Let X be a symplectic homotopy K3 surface and G = S ₅ act on X symplectically. In this paper, we give a weak classification of the G action on X by discussing the fixed-point set structure. Besides, we analyse the exoticness of smooth structures of X under the action of G.