A decomposition theorem for G-groups

Some classical results about linear representations of a finite group G have been also proved for representations of G on non-abelian groups （G-groups）. In this paper we establish a decomposition theorem for irreducible G-groups which expresses a suitable irreducible G-group as a tensor product of two projective G-groups in a similar way to the celebrated theorem of Clifford for linear representations. Moreover, we study the non-abelian minimal normal subgroups of G in which this decomposition is possible.     

Complete reducibility of subgroups of reductive algebraic groups over nonperfect fields III

Let G be a reductive group over a nonperfect field k. We study rationality problems for Serre’s notion of complete reducibility of subgroups of G. In our previous work, we constructed examples of subgroups H of G that are G-completely reducible but not G-completely reducible over k (and vice versa). In this article, we give a theoretical underpinning of those constructions. Then using Geometric Invariant Theory, we obtain a new result on the structure of G(k)-(and G-) orbits in an arbitrary affine G-variety. We discuss several related problems to complement the main results.     

The converse of Schur’s theorem

Schur’s theorem states that for a group G finiteness of G/Z(G) implies the finiteness of G′. In this paper, we show the converse is true provided that G/Z(G) is finitely generated and in such case, we have |G/Z(G)| ≤ |G′| ^d(G/Z(G)). In the special case of G being nilpotent, we prove |G/Z(G)| divides |G′| ^d(G/Z(G)).     

Central Automorphisms and Inner Automorphisms in Finitely Generated Groups

Let G be a group and Aut_c(G) be the group of all central automorphisms of G. We know that in a finite p-group G, Aut_c(G) = Inn(G) if and only if Z(G) = G′ and Z(G) is cyclic. But we shown that we cannot extend this result for infinite groups. In fact, there exist finitely generated nilpotent groups of class 2 in which G′ =Z(G) is infinite cyclic and Inn(G) < C* = Aut_c(G). In this article, we characterize all finitely generated groups G for which the equality Aut_c(G) = Inn(G) holds.     

Mean-field backward stochastic differential equations driven by G-Brownian motion and related partial differential equations

In this paper, we study mean-field backward stochastic differential equations driven by G-Brownian motion (G-BSDEs). We first obtain the existence and uniqueness theorem of these equations. In fact, we can obtain local solutions by constructing Picard contraction mapping for Y term on small interval, and the global solution can be obtained through backward iteration of local solutions. Then, a comparison theorem for this type of mean-field G-BSDE is derived. Furthermore, we establish the connection of this mean-field G-BSDE and a nonlocal partial differential equation. Finally, we give an application of mean-field G-BSDE in stochastic differential utility model.     

Hitting times for random walks on subdivision and triangulation graphs

Let G be a connected graph. The subdivision graph of G, denoted by S(G), is the graph obtained from G by inserting a new vertex into every edge of G. The triangulation graph of G, denoted by R(G), is the graph obtained from G by adding, for each edge uv, a new vertex whose neighbours are u and v. In this paper, we first provide complete information for the eigenvalues and eigenvectors of the probability transition matrix of a random walk on S(G) (res. R(G)) in terms of those of G. Then we give an explicit formula for the expected hitting time between any two vertices of S(G) (res. R(G)) in terms of those of G. Finally, as applications, we show that, the relations between the resistance distances, the number of spanning trees and the multiplicative degree-Kirchhoff index of S(G) (res. R(G)) and G can all be deduced from our results directly.     

Exponent preserving subgroups of the finite simple groups

Given a group G denote with exp(G) its exponent, which is the least common multiple of the order of its elements. In this paper, we solve the problem of finding the finite simple groups having a proper subgroup with the same exponent. For each G with this property we will provide an explicit example of H<G with exp(G)?=?exp(H).     

The spread of the unicyclic graphs

Let G be a simple connected graph with n vertices and n edges which we call a unicyclic graph. In this paper, we first investigate the least eigenvalue λ_n(G), then we present two sharp bounds on the spread s(G) of G.     

Coloring of character graphs

In this paper, we study the character graph Δ(G) of a finite solvable group G. We prove that sum of the chromatic number of Δ(G) and the matching number of complement graph of Δ(G) is equal to the order of Δ(G). Also, we prove that when Δ(G) is not a block, the chromatic number of Δ(G) is equal to the clique number of Δ(G).     

On the character degree sums

Let G be a finite group, and T(G) be the sum of all complex irreducible character degrees of G. In this paper, we get the exact lower bound of |G|∕T(G) for a non-r-solvable group G.     

Prime ideals in crossed products of finite groups

LetR * G be a crossed product of the finite groupG over the ringR. In this paper we discuss the relationship between the prime ideals ofR*G and theG-prime ideals ofR. In particular, we show that Incomparability and Going Down hold in this situation. In the course of the proof, we actually completely describe all the prime idealsP ofR*G such thatP∩R is a fixedG-prime ideal ofR. As an application, we prove that ifG is a finite group of automorphisms ofR, then the prime (primitive) ranks ofR and of the fixed ringR ^G are equal provided •G•⁻∈R. In an appendix, we extend some of these 3 results to crossed products of the infinite cyclic group.     

The nonorientable genus is additive

A graph G is a k-amalgamation of two graphs G₁ and G₂ if G = G₁ ∪ G₂ and G₁ ∩G₂ is a set of k vertices. In this paper we show that γ(G) differs from γ(G₁) + γ(G₂) by at most a quadratic on k, where γ denotes the nonorientable genus of a graph. In the sequel to this paper we show that no such bound holds for the orientable genus of k-amalgamations.     

On graphs determining links with maximal number of components via medial construction

Let G be a connected plane graph, D(G) be the corresponding link diagram via medial construction, and μ(D(G)) be the number of components of the link diagram D(G). In this paper, we first provide an elementary proof that μ(D(G))≤n(G)+1, where n(G) is the nullity of G. Then we lay emphasis on the extremal graphs, i.e. the graphs with μ(D(G))=n(G)+1. An algorithm is given firstly to judge whether a graph is extremal or not, then we prove that all extremal graphs can be obtained from K₁ by applying two graph operations repeatedly. We also present a dual characterization of extremal graphs and finally we provide a simple criterion on structures of bridgeless extremal graphs.     

Autocommuting probability of a finite group

Let G be a finite group and Aut(G) the automorphism group of G. The autocommuting probability of G, denoted by Pr(G,Aut(G)), is the probability that a randomly chosen automorphism of G fixes a randomly chosen element of G. In this paper, we study Pr(G,Aut(G)) through a generalization. We obtain a computing formula, several bounds and characterizations of G through Pr(G,Aut(G)). We conclude the paper by showing that the generalized autocommuting probability of G remains unchanged under autoisoclinism.     

Deformation of proper actions on reductive homogeneous spaces

Let G be a real reductive Lie group and H a closed reductive subgroup of G. We investigate the deformation of standard compact quotients of G/H, that is, of quotients of G/H by discrete groups Γ that are uniform lattices in some closed reductive subgroup L of G acting properly and cocompactly on G/H. For L of real rank 1, we prove that after a small deformation in G, such a group Γ keeps acting properly discontinuously and cocompactly on G/H. More generally, we prove that the properness of the action of any convex cocompact subgroup of L on G/H is preserved under small deformations, and we extend this result to reductive homogeneous spaces G/H over any local field. As an application, we obtain compact quotients of SO(2n, 2)/U(n, 1) by Zariski-dense discrete subgroups of SO(2n, 2) acting properly discontinuously.     

Claw‐free circular‐perfect graphs

The circular chromatic number of a graph is a well‐studied refinement of the chromatic number. Circular‐perfect graphs form a superclass of perfect graphs defined by means of this more general coloring concept. This article studies claw‐free circular‐perfect graphs. First, we prove that if G is a connected claw‐free circular‐perfect graph with χ(G)>ω(G), then min{α(G), ω(G)}=2. We use this result to design a polynomial time algorithm that computes the circular chromatic number of claw‐free circular‐perfect graphs. A consequence of the strong perfect graph theorem is that minimal imperfect graphs G have min{α(G), ω(G)}=2. In contrast to this result, it is shown in Z. Pan and X. Zhu [European J Combin 29(4) (2008), 1055–1063] that minimal circular‐imperfect graphs G can have arbitrarily large independence number and arbitrarily large clique number. In this article, we prove that claw‐free minimal circular‐imperfect graphs G have min{α(G), ω(G)}≤3. © 2010 Wiley Periodicals, Inc. J Graph Theory 65: 163–172, 2010     

A note on unimodular congruence of the Laplacian matrix of a graph

Let G be a simple connected graph and L(G) be its Laplacian matrix. In this note, we prove that L(G) is congruent by a unimodular matrix to its Smith normal form if and only if G is a tree.     

On chromatic‐choosable graphs

A graph is chromatic‐choosable if its choice number coincides with its chromatic number. It is shown in this article that, for any graph G, if we join a sufficiently large complete graph to G, then we obtain a chromatic‐choosable graph. As a consequence, if the chromatic number of a graph G is close enough to the number of vertices in G, then G is chromatic‐choosable. We also propose a conjecture related to this fact. © 2002 Wiley Periodicals, Inc. J Graph Theory 40: 130–135, 2002     

A note on <Emphasis Type="Italic">p</Emphasis>-groups of order ≤ <Emphasis Type="Italic">p</Emphasis><Superscript>4</Superscript>

In [1], we defined c(G), q(G) and p(G). In this paper we will show that if G is a p-group, where p is an odd prime and |G| ≤ p ⁴, then c(G) = q(G) = p(G). However, the question of whether or not there is a p-group G with strict inequality c(G) = q(G) < p(G) is still open.     

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.