On the Surjectivity of the Exponential Function of Solvable Lie Groups

For a solvable Lie group G the surjectivity of the exponential function exp_G is equivalent to the connectedness of the near-Cartan subgroups and to the connectedness of the centralizers in a Cartan subgroup of all nilpotent elements in its Lie algebra g. Furthermore, these conditions are satisfied if and only if for all elements g ? G there is an x ? g with g = exp_G x in which exp_G is regular.     

Groups whose nonlinear irreducible characters separate element orders or conjugacy class sizes

A class function φ on a finite group G is said to be an order separator if, for every x and y in G \ {1}, φ(x) = φ(y) is equivalent to x and y being of the same order. Similarly, φ is said to be a class-size separator if, for every x and y in G\ {1}, φ(x) = φ(y) is equivalent to |C _G (x)| = |C _G (y)|. In this paper, finite groups whose nonlinear irreducible complex characters are all order separators (respectively, class-size separators) are classified. In fact, a more general setting is studied, from which these classifications follow. This analysis has some connections with the study of finite groups such that every two elements lying in distinct conjugacy classes have distinct orders, or, respectively, in which disctinct conjugacy classes have distinct sizes. Received: 10 April 2007     

The finite simple groups with at most two fusion classes of every order

Elements a,b of a group G are said to be fused if a = b^σ and to be inverse-fused if a =(b^-1)^σ for some σ ? Aut(G). The fusion class of a ? G is the set {a^σ | σ ? Aut(G)}, and it is called a fusion class of order i if a has order iThis paper gives a complete classification of the finite nonabelian simple groups G for which either (i) or (ii) holds, where:

(i) G has at most two fusion classes of order i for every i (23 examples); and

(ii) any two elements of G of the same order are fused or inversenfused.

The examples in case (ii) are: A₅, A₆,L₂(7),L₂(8), L₃(4), Sz(8), M₁₁ and M₂₃An application is given concerning isomorphisms of Cay ley graphs.     

Monochrome symmetric subsets of colored groups

In (Electron. J. Combin. 10 (2003); http://www.combinatorics.org/volume-10/Abstracts/v1oi1r28.html), the first author (Yuliya Gryshko) asked three questions. Is it true that every infinite group admitting a 2-coloring without infinite monochromatic symmetric subsets is either almost cyclic (i.e., have a finite index subgroup which is cyclic infinite) or countable locally finite? Does every infinite group G include a monochromatic symmetric subset of any cardinal <|G| for any finite coloring? Does every uncountable group G such that |B(G)|< |G| where B(G)={xG:x²=1}, admit a 2-coloring without monochromatic symmetric subsets of cardinality |G|? We answer the first question positively. Assuming the generalized continuum hypothesis (GCH), we give a positive answer to the second question in the abelian case. Finally, we build a counter-example for the third question and we give a necessary and sufficient condition for an infinite group G to admit 2-coloring without monochromatic symmetric subsets of cardinality |G|. This generalizes some results of Protasov on infinite abelian groups (Mat. Zametki 59 (1996) 468–471; Dopovidi NAN Ukrain 1 (1999) 54–57).     

On Solvability of Finite Groups with Few Non-Normal Subgroups

For a finite group G, let v(G) denote the number of conjugacy classes of non-normal subgroups of G and vc(G) denote the number of conjugacy classes of non-normal noncyclic subgroups of G. In this paper, we show that every finite group G satisfying v(G) ≤2|π(G)| or vc(G) ≤ |π(G)| is solvable, and for a finite nonsolvable group G, v(G) = 2|π(G)| +1 if and only if G ? A ₅.     

Essential subgroups of topological groups

We study the class Wof Hausdorff topological groups Gfor which the following two cardinal invariants coincide

ES(G)=min{|H|:H≤ Gdense and essential}

TD(G)=min{|H|:H≤ Gtotally dense}

We prove that W contains the following classes:locally compact abelian groups, compact connected groups, countably compact totally discon¬nected abelian groups, topologically simple groups, locally compact Abelian groups when endowed with their Bohr topology, totally minimal abelian groups and free Abelian topological groups. For all these classes we are also able to giv ean explicit computation of the common value of ESand TD.     

On a Commuting Graph on Conjugacy Classes of Groups

We consider the graph Γ(G), associated with the conjugacy classes of a group G. Its vertices are the nontrivial conjugacy classes of G, and we join two different classes C, D, whenever there exist x ∈ G and y ∈ D such that xy = yx. The aim of this article is twofold. First, we investigate which graphs can occur in various contexts and second, given a graph Γ(G) associated with G, we investigate the possible structure of G. We proved that if G is a periodic solvable group, then Γ(G) has at most two components, each of diameter at most 9. If G is any locally finite group, then Γ(G) has at most 6 components, each of diameter at most 19. Finally, we investigated periodic groups G with Γ(G) satisfying one of the following properties: (i) no edges exist between noncentral conjugacy classes, and (ii) no edges exist between infinite conjugacy classes. In particular, we showed that the only nonabelian groups satisfying (i) are the three finite groups of order 6 and 8.     

Some Properties of Bell Groups

For any integer n ≠ 0,1, a group G is said to be “n-Bell” if it satisfies the identity [x ⁿ ,y] = [x,y ⁿ ]. It is known that if G is an n-Bell group, then the factor group G/Z ₂(G) has finite exponent dividing 12n ⁵(n ? 1)⁵. In this article we show that this bound can be improved. Moreover, we prove that every n-Bell group is n-nilpotent; consequently, using results of Baer on finite n-nilpotent groups, we give the structure of locally finite n-Bell groups. Finally, we are concerned with locally graded n-Bell groups for special values of n.     

Non-Commuting Graphs of Nilpotent Groups

Let G be a non-abelian group and Z(G) be the center of G. The non-commuting graph Γ _G associated to G is the graph whose vertex set is G?Z(G) and two distinct elements x, y are adjacent if and only if xy ≠ yx. We prove that if G and H are non-abelian nilpotent groups with irregular isomorphic non-commuting graphs, then |G| = |H|.     

Finite Groups all of Whose Second Maximal Subgroups are QTI-Subgroups

A subgroup H of a finite group G is called a QTI-subgroup if C _G (x) ≤ N _G (H) for any 1 ≠ x ∈ H. In this article, the finite groups all of whose second maximal subgroup are QTI-subgroups are classified.     

A Note on Almost Regular Group Rings

It is proved that a group ring R = AG is almost regular if and only if (i) the ring A is almost regular; (ii) the group G is locally finite; (iii) the order |H| of every finite subgroup H of G is invertible in A. Bibliography: 7 titles.     

On finite groups with non-subnormal subgroups

For a finite group G, let n(G) denote the number of conjugacy classes of non-subnormal subgroups of G. In this paper, we show that a finite group G satisfying n(G)≤2|π(G)| is solvable, and for a finite non-solvable group G, n(G) = 2|π(G)|+1 if and only if G?A₅.     

Noncyclic Graph of a Group

We associate a graph Γ _G to a nonlocally cyclic group G (called the noncyclic graph of G) as follows: take G\ Cyc(G) as vertex set, where Cyc(G) = {x ? G| 〈x, y〉 is cyclic for all y ? G}, and join two vertices if they do not generate a cyclic subgroup. We study the properties of this graph and we establish some graph theoretical properties (such as regularity) of this graph in terms of the group ones. We prove that the clique number of Γ _G is finite if and only if Γ _G has no infinite clique. We prove that if G is a finite nilpotent group and H is a group with Γ _G  ? Γ _H and |Cyc(G)| = |Cyc(H)| = 1, then H is a finite nilpotent group. We give some examples of groups G whose noncyclic graphs are “unique”, i.e., if Γ _G  ? Γ _H for some group H, then G ? H. In view of these examples, we conjecture that every finite nonabelian simple group has a unique noncyclic graph. Also we give some examples of finite noncyclic groups G with the property that if Γ _G  ? Γ _H for some group H, then |G| = |H|. These suggest the question whether the latter property holds for all finite noncyclic groups.     

An improvement of fraisse's sufficient condition for hamiltonian graphs

Let G be a k-connected graph of order n. For an independent set c, let d(S) be the number of vertices adjacent to at least one vertex of S and > let i(S) be the number of vertices adjacent to at least |S| vertices of S. We prove that if there exists some s, 1 ≤ s ≤ k, such that Σ_xiEX d(X\{X_i}) > s(n?1) – k[s/2] – i(X)[(s?1)/2] holds for every independetn set X ={x₀, x₁ ?x_s} of s + 1 vertices, then G is hamiltonian. Several known results, including Fraisse's sufficient condition for hamiltonian graphs, are dervied as corollaries.     

On ℳ p -Supplemented Subgroups of Finite Groups

A subgroup H of G is called ? _p -supplemented in G if there exists a subgroup B of G such that G = HB and TB < G for every maximal subgroup T of H with |H: T| =p ^α. In this paper, we investigate the influence of ? _p -supplemented subgroup and some conditions for p-nilpotency and p-supersolvability of finite groups are obtained.     

On Groups with a CC(n)-Subgroup

A subgroup H of G is a CC(n)-subgroup of G if |G: H| >n and |C_G(x): C_H(x)| ≤n for each element x ∈ H ? {1}. In this article, we study the finite p-groups with a nontrivial CC(p)-subgroup, and the locally nilpotent groups with a nontrivial CC(n)-subgroup.     

Adian-Lisenok groups and (U) condition

A group G possesses the property (U) with respect to S if there exists a number M = M(G) such that for each generating set P of the group G there exists an element t ? G for which max _x?S |t ^?1 xt| _P ≤ M. It is proved that the well-known Adian-Lisenok groups possess the property (U). In connection with the problem on finding infinite groups with the property (U), which is stated in a joint unpublishedwork by D.Osin and D. Sonkin, it is shown that for any odd n ≥ 1003 there is a continuum set of non-isomorphic, i.e. simple groups with the property (U) in the variety of groups satisfying the identity x ⁿ = 1.     

Connected subgraphs with small degree sums in 3‐connected planar graphs

It is well‐known that every planar graph has a vertex of degree at most five. Kotzig proved that every 3‐connected planar graph has an edge xy such that deg(x) + deg (y) ≤ 13. In this article, considering a similar problem for the case of three or more vertices that induce a connected subgraph, we show that, for a given positive integer t, every 3‐connected planar graph G with |V(G)| ≥ t has a connected subgraph H of order t such that Σ_x∈V(H) deg_G(x) ≤ 8t − 1. As a tool for proving this result, we consider decompositions of 3‐connected planar graphs into connected subgraphs of order at least t and at most 2t − 1. © 1999 John Wiley & Sons, Inc. J Graph Theory 30: 191–203, 1999     

Automorphisms of Hyperbolic Groups and Graphs of Groups

Using the canonical JSJ splitting, we describe the outer automorphism group Out(G) of a one-ended word hyperbolic group G. In particular, we discuss to what extent Out(G) is virtually a direct product of mapping class groups and a free abelian group, and we determine for which groups Out(G) is infinite. We also show that there are only finitely many conjugacy classes of torsion elements in Out(G), for G any torsion-free hyperbolic group. More generally, let Γ be a finite graph of groups decomposition of an arbitrary group G such that edge groups G_e are rigid (i.e. Out(G_e) is finite). We describe the group of automorphisms of G preserving Γ, by comparing it to direct products of suitably defined mapping class groups of vertex groups.     

Existence of subgraph with orthogonal (g,f)-factorization

Simple graphs are considered. Let G be a graph andg(x) andf(x) integer-valued functions defined on V(G) withg(x)⩽f(x) for everyxɛV(G). For a subgraphH ofG and a factorizationF=|F ₁,F ₂,⃛,F ₁| ofG, if |E(H)∩E(F ₁)|=1,1⩽i⩽j, then we say thatF orthogonal toH. It is proved that for an (mg(x)+k,mf(x) -k)-graphG, there exists a subgraphR ofG such that for any subgraphH ofG with |E(H)|=k,R has a (g,f)-factorization orthogonal toH, where 1⩽k<m andg(x)⩾1 orf(x)⩾5 for everyxɛV(G). Project supported by the Chitia Postdoctoral Science Foundation and Chuang Xin Foundation of the Chinese Academy of Sciences.