Binding number and minimum degree for the existence of (g,f,n)-critical graphs

Let G be a graph of order p. The binding number of G is defined as $\mbox{bind}(G):=\min\{\frac{|N_{G}(X)|}{|X|}\mid\emptyset\neq X\subseteq V(G)\,\,\mbox{and}\,\,N_{G}(X)\neq V(G)\}$ . Let g(x) and f(x) be two nonnegative integer-valued functions defined on V(G) with g(x)≤f(x) for any x∈V(G). A graph G is said to be (g,f,n)-critical if G?N has a (g,f)-factor for each N?V(G) with |N|=n. If g(x)≡a and f(x)≡b for all x∈V(G), then a (g,f,n)-critical graph is an (a,b,n)-critical graph. In this paper, several sufficient conditions on binding number and minimum degree for graphs to be (a,b,n)-critical or (g,f,n)-critical are given. Moreover, we show that the results in this paper are best possible in some sense.     

On the extendability of Bi-Cayley graphs of finite abelian groups

Let G be a finite group and A a nonempty subset (possibly containing the identity element) of G. The Bi-Cayley graph X=BC(G,A) of G with respect to A is defined as the bipartite graph with vertex set G×{0,1} and edge set {{(g,0),(sg,1)}∣g∈G,s∈A}. A graph Γ admitting a perfect matching is called n-extendable if ∣V(Γ)∣≥2n+2 and every matching of size n in Γ can be extended to a perfect matching of Γ. In this paper, the extendability of Bi-Cayley graphs of finite abelian groups is explored. In particular, 2-extendable and 3-extendable Bi-Cayley graphs of finite abelian groups are characterized.     

Nil-automorphisms of groups with residual properties

Let G be any group and x an automorphism of G. The automorphism x is said to be nil if, for every g ∈ G, there exists n = n(g) such that [g, _n x] = 1. If n can be chosen independently of g, we say that x is n-unipotent. A nil (resp. unipotent) automorphism x could also be seen as a left Engel element (resp. left n-Engel element) in the group G〈x〉. When G is a finite dimensional vector space, groups of unipotent linear automorphisms turn out to be nilpotent, so that one might ask to what extent this result can be extended to a more general setting. In this paper we study finitely generated groups of nil or unipotent automorphisms of groups with residual properties (e.g. locally graded groups, residually finite groups, profinite groups), proving that such groups are nilpotent.     

Characteristic polynomials of a permutation representation

Let G be a finite group. To each permutation representation (G, X) of G and class function χ of G we associate the χ-characteristic polynomialg_χ(x) of (G, X) which is a polynomial in one variable with complex numbers as coefficients. The coefficients of g_χ(x) are investigated in terms of the “exterior powers” of (G, X). If χ is the principal character 1_G of G, the coefficients of g_χ(x) are non-negative integers; and if furthermore G has odd order, the kth coefficient is the number of orbits of G acting on the subsets of X with k elements. Quadratic and linear relations among the exterior powers of (G, X) have been derived and it is shown how the polynomial g_χ(x) can be computed from the cycle index of (G, X).     

On the multiple exterior degree of finite groups

Recently the first two authors have introduced a group invariant, called exterior degree, which is related to the number of elements x and y of a finite group G such that xΛy = 1 in the exterior square GΛG of G. Research on this topic gives some relations between this concept, the Schur multiplier and the capability of a finite group. In the present paper, we will generalize the concept of exterior degree of groups and we will introduce the multiple exterior degree of finite groups. Among other results, we will obtain some relations between the multiple exterior degree, multiple commutativity degree and capability of finite groups.     

Graphs of nonsolvable groups with four degree-vertices

Let G be a finite group. The degree(vertex) graph Γ(G) attached to G is a character degree graph.Its vertices are the degrees of the nonlinear irreducible complex characters of G, and different vertices m, n are adjacent if the greatest common divisor(m, n) 1. In this paper, we classify all graphs with four vertices that occur as Γ(G) for nonsolvable groups G.     

Vertex pancyclicity and new sufficient conditions

For a graph G, ??(G) denotes the minimum degree of G. In 1971, Bondy proved that, if G is a 2-connected graph of order n and d(x)?+?d(y)????n for each pair of non-adjacent vertices x,y in G, then G is pancyclic or G?=?K _n/2,n/2. In 2001, Xu proved that, if G is a 2-connected graph of order n????6 and |N(x)????N(y)|?+???(G)????n for each pair of non-adjacent vertices x,y in G, then G is pancyclic or G?=?K _n/2,n/2. In this paper, we introduce a new sufficient condition of generalizing degree sum and neighborhood union and prove that, if G is a 2-connected graph of order n????6 and |N(x)????N(y)|?+?d(w)????n for any three vertices x,y,w of d(x,y)?=?2 and wx or $wy\not\in E(G)$ in G, then G is 4-vertex pancyclic or G belongs to two classes of well-structured exceptional graphs. This result also generalizes the above results.     

On a conjecture of E. Rapaport Strasser about knot-like groups and its pro-p version

A group G is knot-like if it is finitely presented of deficiency 1 and has abelianization G/G^′?Z. We prove the conjecture of E. Rapaport Strasser that if a knot-like group G has a finitely generated commutator subgroup G^′ then G^′ should be free in the special case when the commutator G^′ is residually finite. It is a corollary of a much more general result : if G is a discrete group of geometric dimension n with a finite K(G,1)-complex Y of dimension n, Y has Euler characteristics 0, N is a normal residually finite subgroup of G, N is of homological type FP_n-1 and G/N?Z then N is of homological type FP_n and hence G/N has finite virtual cohomological dimension vcd(G/N)=cd(G)-cd(N). In particular either N has finite index in G or cd(N)?cd(G)-1.Furthermore we show a pro-p version of the above result with the weaker assumption that G/N is a pro-p group of finite rank. Consequently a pro-p version of Rapaport's conjecture holds.     

Arithmetical conditions on the conjugacy vector of a finite group

We get new properties of the numbersr _G(xN) = |{Cl _G (g)|Cl _G (g)∩xN ≠ Ø} (whereG is a finite group andN is a normal subgroup ofG) that are useful in the analysis of the classification of the finite groups according to the number of conjugacy classes.     

On the homotopy type of CW-complexes with aspherical fundamental group

This paper is concerned with the homotopy type distinction of finite CW-complexes. A (G,n)-complex is a finite n-dimensional CW-complex with fundamental-group G and vanishing higher homotopy-groups up to dimension n−1. In case G is an n-dimensional group there is a unique (up to homotopy) (G,n)-complex on the minimal Euler-characteristic level χmin(G,n). For every n we give examples of n-dimensional groups G for which there exist homotopically distinct (G,n)-complexes on the level χmin(G,n)+1. In the case where n=2 these examples are algebraic.     

Assouad-Nagata dimension of locally finite groups and asymptotic cones

In this paper we study two problems concerning Assouad-Nagata dimension:

(1)

Is there a metric space of positive asymptotic Assouad-Nagata dimension such that all of its asymptotic cones are of Assouad-Nagata dimension zero? (Dydak and Higes, 2008 [11, Question 4.5]).

(2)

Suppose G is a locally finite group with a proper left invariant metric dG. If dimAN(G,dG)>0, is dimAN(G,dG) infinite? (Brodskiy et al., preprint [6, Problem 5.3]).

The first question is answered positively. We provide examples of metric spaces of positive (even infinite) Assouad-Nagata dimension such that all of its asymptotic cones are ultrametric. The metric spaces can be groups with proper left invariant metrics.The second question has a negative solution. We show that for each n there exists a locally finite group of Assouad-Nagata dimension n. As a consequence this solves for non-finitely generated countable groups the question about the existence of metric spaces of finite asymptotic dimension whose asymptotic Assouad-Nagata dimension is larger but finite.     

On Finite n-Acentralizer Groups

Let G be a group and Aut(G) be the group of automorphisms of G. Then the Acentralizer of an automorphism α ∈Aut(G) in G is defined as C _G (α) = {g ∈ G∣α(g) = g}. For a finite group G, let Acent(G) = {C _G (α)∣α ∈Aut(G)}. Then for any natural number n, we say that G is n-Acentralizer group if |Acent(G)| =n. We show that for any natural number n, there exists a finite n-Acentralizer group and determine the structure of finite n-Acentralizer groups for n ≤ 5.     

Path-Star Ramsey numbers

Let f(m,n) be the least integer N such that for every graph G with N vertices, either G contains a path of m vertices or the complement of G contains a vertex of degree at least n. This paper determines f(m,n) for all m, n.     

Variations on a Theme of Cline and Donkin

Let N be a normal subgroup of a group G. An N-module Q is called G-stable provided that Q is equivalent to the twist Q ^g of Q by g, for every g?∈?G. If the action of N on Q extends to an action of G on Q, then Q is obviously G-stable, but the converse need not hold. A famous conjecture in the modular representation theory of reductive algebraic groups G asserts that the (obviously G-stable) projective indecomposable modules (PIMs) Q for the Frobenius kernels G _r (r?≥?1) of G have a G-module structure. It is sometimes just as useful (for a general module Q) to know that a finite direct sum Q ^?⊕?n of Q has a compatible G-module structure. In this paper, this property is called numerical stability. In recent work (Parshall and Scott, Adv Math 226:2065–2088, 2011), the authors established numerical stability in the special case of PIMs. We provide in this paper a more general context for that result, working in the context of k-group schemes and a suitable version of G-stability, called strong G-stability. Among our results here is the determination of necessary and sufficient conditions for the existence of a compatible G-module structure on a strongly G-stable N-module, in the form of a cohomological obstruction which must be trivial precisely when the G-module structure exists. Our main result is achieved by giving an approach to killing the obstruction by tensoring with certain finite dimensional G/N-modules.     

Coloring graphs from random lists of size 2

Let G=G(n) be a graph on n vertices with girth at least g and maximum degree bounded by some absolute constant Δ. Assign to each vertex v of G a list L(v) of colors by choosing each list independently and uniformly at random from all 2-subsets of a color set C of size σ(n). In this paper we determine, for each fixed g and growing n, the asymptotic probability of the existence of a proper coloring φ such that φ(v)∈L(v) for all v∈V(G). In particular, we show that if g is odd and σ(n)=ω(n^1/(2g−2)), then the probability that G has a proper coloring from such a random list assignment tends to 1 as n→∞. Furthermore, we show that this is best possible in the sense that for each fixed odd g and each n≥g, there is a graph H=H(n,g) with bounded maximum degree and girth g, such that if σ(n)=o(n^1/(2g−2)), then the probability that H has a proper coloring from such a random list assignment tends to 0 as n→∞. A corresponding result for graphs with bounded maximum degree and even girth is also given. Finally, by contrast, we show that for a complete graph on n vertices, the property of being colorable from random lists of size 2, where the lists are chosen uniformly at random from a color set of size σ(n), exhibits a sharp threshold at σ(n)=2n.     

Irreducible linear subgroups generated by pairs of matrices with large irreducible submodules

We call an element of a finite general linear group GL(d, q) fat if it leaves invariant and acts irreducibly on a subspace of dimension greater than d/2. Fatness of an element can be decided efficiently in practice by testing whether its characteristic polynomial has an irreducible factor of degree greater than d/2. We show that for groups G with SL(d, q) ≤ G ≤ GL(d, q) most pairs of fat elements from G generate irreducible subgroups, namely we prove that the proportion of pairs of fat elements generating a reducible subgroup, in the set of all pairs in G × G, is less than q ^−d+1. We also prove that the conditional probability to obtain a pair (g ₁, g ₂) in G × G which generates a reducible subgroup, given that g ₁, g ₂ are fat elements, is less than 2q ^−d+1. Further, we show that any reducible subgroup generated by a pair of fat elements acts irreducibly on a subspace of dimension greater than d/2, and in the induced action the generating pair corresponds to a pair of fat elements.     

Free actions on handlebodies

The equivalence (or weak equivalence) classes of orientation-preserving free actions of a finite group G on an orientable three-dimensional handlebody of genus g?1 can be enumerated in terms of sets of generators of G. They correspond to the equivalence classes of generating n-vectors of elements of G, where n=1+(g−1)/|G|, under Nielsen equivalence (or weak Nielsen equivalence). For Abelian and dihedral G, this allows a complete determination of the equivalence and weak equivalence classes of actions for all genera. Additional information is obtained for other classes of groups. For all G, there is only one equivalence class of actions on the genus g handlebody if g is at least 1+?(G)|G|, where ?(G) is the maximal length of a chain of subgroups of G. There is a stabilization process that sends an equivalence class of actions to an equivalence class of actions on a higher genus, and some results about its effects are obtained.     

Characterization of almost maximally almost-periodic groups

Let G be an Abelian group. We prove that a group G admits a Hausdorff group topology τ such that the von Neumann radical n(G,τ) of (G,τ) is non-trivial and finite iff G has a non-trivial finite subgroup. If G is a topological group, then n(n(G))≠n(G) if and only if n(G) is not dually embedded. In particular, n(n(Z,τ))=n(Z,τ) for any Hausdorff group topology τ on Z.     

On finite groups with exactly seven element centralizers

For a finite groupG, #Cent(G) denotes the number of centralizers of its elements. A groupG is calledn-centralizer if #Cent(G) =n, and primitive n-centralizer if $\# Cent(G) = \# Cent\left( {\frac{G}{{Z(G)}}} \right) = n$ . The first author in [1], characterized the primitive 6-centralizer finite groups. In this paper we continue this problem and characterize the primitive 7-centralizer finite groups. We prove that a finite groupG is primitive 7-centralizer if and only if $\frac{G}{{Z(G)}} \cong D_{10} $ orR, whereR is the semidirect product of a cyclic group of order 5 by a cyclic group of order 4 acting faithfully. Also, we compute#Cent(G) for some finite groups, using the structure ofG modulu its center.     

On finite groups with a certain number of centralizers

LetG be a finite group and #Cent(G) denote the number of centralizers of its elements.G is calledn-centralizer if #Cent(G)=n, and primitiven-centralizer if #Cent(G)=#Cent(G/Z(G))=n. In this paper we investigate the structure of finite groups with at most 21 element centralizers. We prove that such a group is solvable and ifG is a finite group such thatG/Z(G)?A₅, then #Cent(G)=22 or 32. Moroever, we prove that A₅ is the only finite simple group with 22 centralizers. Therefore we obtain a characterization of A₅ in terms of the number of centralizers