On Groups with the Second Largest Value of the Sum of Element Orders

For a finite group G, let ψ(G) denote the sum of element orders of G. It is known that the maximum value of ψ on the set of groups of order n, where n is a positive integer, will occur at the cyclic group ? _n . In this paper, we investigate groups with the second largest value of ψ on the set of groups of the same order.     

On certain subgraphs of a complete transitively directed graph

Let G_n be a complete transitively directed graph with n + 1 vertices v₀, v₁, …, v_n. Let ψ(n) be the number of subgraphs H of G_n where each vertex in H lies along a directed path from v₀ to v_n in H. ψ(n) and some related quantities are calculated.     

On graph invariants satisfying the deletion-contraction formula

As a generalization of chromatic polynomials, this paper deals with real-valued mappings ψ on the class of graphs satisfying ψ(G₁) = ψ(G₂) for all pairs G₁, G₂ of isomorphic graphs and ψ(G) = ψ(G − e) − ψ(G/e) for all graphs G and all edges e of G, where the definition of G/e is nonstandard. In particular, new inequalities for chromatic polynomials are presented. © 1996 John Wiley & Sons, Inc.     

On the description of overgroups of E(m,R)?E(n,R)

The subgroups E(m,R) ⊗ E(n,R) ≤ H ≤ G = GL(mn,R) are studied under the assumption that the ring R is commutative and m, n ≥ 3. The group GL _m ⊗GL _n is defined by equations, the normalizer of the group E(m,R) ⊗ E(n,R) is calculated, and with each intermediate subgroup H it is associated a uniquely determined lower level (A,B,C), where A,B,C are ideals in R such that mA,A ² ≤ B ≤ A and nA,A ² ≤ C ≤ A. The lower level specifies the largest elementary subgroup satisfying the condition E(m, n,R, A,B,C) ≤ H. The standard answer to this problem asserts that H is contained in the normalizer N _G (E(m,n,R, A,B,C)). Bibliography: 46 titles.     

On Group Chromatic Number of Graphs

Let G be a graph and A an Abelian group. Denote by F(G, A) the set of all functions from E(G) to A. Denote by D an orientation of E(G). For f ∈ F(G,A), an (A,f)-coloring of G under the orientation D is a function c : V(G)↦A such that for every directed edge uv from u to v, c(u)−c(v) ≠ f(uv). G is A-colorable under the orientation D if for any function f ∈ F(G, A), G has an (A, f)-coloring. It is known that A-colorability is independent of the choice of the orientation. The group chromatic number of a graph G is defined to be the least positive integer m for which G is A-colorable for any Abelian group A of order ≥m, and is denoted by χ_g(G). In this note we will prove the following results. (1) Let H₁ and H₂ be two subgraphs of G such that V(H₁)∩V(H₂)=∅ and V(H₁)∪V(H₂)=V(G). Then χ_g(G)≤min{max{χ_g(H₁), max_v∈V(H2)deg(v,G)+1},max{χ_g(H₂), max_u∈V(H1) deg (u, G) + 1}}. We also show that this bound is best possible. (2) If G is a simple graph without a K_3,3-minor, then χ_g(G)≤5.     

θ-Pairs for 2-maximal subgroups of finite groups

Let H, A and B be subgroups of a group G. We call the pair (A, B) a θ-pair for H in G if: (i) áH, A? = G{\langle H, A\rangle=G} and B = (A ∩ H) _G ; (ii) if A ₁/B is a proper subgroup of A/B and A₁/B \vartriangleleft G/B{{A_1/B \vartriangleleft G/B}}, then G 1 áH, A₁?{G\neq \langle H, A_1\rangle}. In this paper, we study the θ-pairs for 2-maximal subgroups of a group, which imply a group to be solvable or supersolvable.     

On Finite Groups with a Given Number of Centralizers

For a finite group G, let Cent(G) denote the set of centralizers of single elements of G and #Cent(G) = |Cent(G)|. G is called an n-centralizer group if #Cent(G) = n, and a primitive n-centralizer group if #Cent(G) = #Cent(G/Z(G)) = n. In this paper, we compute #Cent(G) for some finite groups G and prove that, for any positive integer n 2, 3, there exists a finite group G with #Cent(G) = n, which is a question raised by Belcastro and Sherman [2]. We investigate the structure of finite groups G with #Cent(G) = 6 and prove that, if G is a primitive 6-centralizer group, then G/Z(G) A₄, the alternating group on four letters. Also, we prove that, if G/Z(G) A₄, then #Cent(G) = 6 or 8, and construct a group G with G/Z(G) A₄ and #Cent(G) = 8.This research was in part supported by a grant from IPM.2000 Mathematics Subject Classification: 20D99, 20E07     

A Note on Projective and Flat Dimensions and the Bieri-Neumann-Strebel-Renz Σ-Invariants

Let G be a finitely generated group, and A a ?[G]-module of flat dimension n such that the homological invariant Σ ⁿ (G, A) is not empty. We show that A has projective dimension n as a ?[G]-module. In particular, if G is a group of homological dimension hd(G) = n such that the homological invariant Σ ⁿ (G, ?) is not empty, then G has cohomological dimension cd(G) = n. We show that if G is a finitely generated soluble group, the converse is true subject to taking a subgroup of finite index, i.e., the equality cd (G) = hd(G) implies that there is a subgroup H of finite index in G such that Σ^∞(H, ?) ≠ ?.     

On self-normality and abnormality in the alternating groups

It is known that in a finite solvable group G, a subgroup H is abnormal if and only if every subgroup of G containing H is self-normalizing in G. Although, in general, the assumption of solvability cannot be dropped, in this paper we prove the theorem for the special case G = A_n and H a second maximal intransitive subgroup of A_n.Received: 1 July 2003     

Graph norms and Sidorenko’s conjecture

Let H and G be two finite graphs. Define h _H (G) to be the number of homomorphisms from H to G. The function h _H (·) extends in a natural way to a function from the set of symmetric matrices to ℝ such that for A _G , the adjacency matrix of a graph G, we have h _H (A _G ) = h _H (G). Let m be the number of edges of H. It is easy to see that when H is the cycle of length 2n, then h _H (·)^1/m is the 2n-th Schatten-von Neumann norm. We investigate a question of Lovász that asks for a characterization of graphs H for which the function h _H (·)^1/m is a norm.     

On finite groups whose every proper normal subgroup is a union of a given number of conjugacy classes

Let G be a finite group andA be a normal subgroup ofG. We denote by ncc(A) the number ofG-conjugacy classes ofA andA is calledn-decomposable, if ncc(A)= n. SetK _G = {ncc(A)|A ⊲ G}. LetX be a non-empty subset of positive integers. A groupG is calledX-decomposable, ifK _G =X. Ashrafi and his co-authors [1-5] have characterized theX-decomposable non-perfect finite groups forX = {1, n} andn ≤ 10. In this paper, we continue this problem and investigate the structure ofX-decomposable non-perfect finite groups, forX = {1, 2, 3}. We prove that such a group is isomorphic to Z₆, D₈, Q₈, S₄, SmallGroup(20, 3), SmallGroup(24, 3), where SmallGroup(m, n) denotes the mth group of ordern in the small group library of GAP [11].     

Symmetric cohomology of groups in low dimension

We give an explicit characterization for group extensions that correspond to elements of the symmetric cohomology HS ²(G, A). We also give conditions for the map HS ⁿ (G, A) → H ⁿ (G, A) to be injective.     

Some nordhaus-- gaddum-type results

A Nordhaus--Gaddum-type result is a (tgiht) lower or upper bound on the sum or product of a parameter of a graph and its complement. In this paper some variations are considered. First, the sums and products of ψ(G₁) and ψ(G₂) are examined where G₁ ⊕ G₂ = K(s, s), and ψ is the independence, domination, or independent domination number, inter alia. In particular, it is shown that the maximum value of the product of the domination numbers of G₁ and G₂ is [(s/2 + 2)²] for s ≥ 3. Thereafter it is shown that for H₁ ⊕ H₂ ⊕ H₃ = K_p, the maximum product of the domination numbers of H₂, H₂, and H₃ is p³/27 + Θ(p²).     

Matrix Near-rings over Centralizer Near-rings

For a finite group G and a subgroup A of Aut(G), let M_A(G) denote the centralizer near-ring determined by A and G. The group G is an M_A(G)-module. Using the action of M_A(G) on G, one has the n × n generalized matrix near-ring Mat_n(M_A(G);G). The correspondence between the ideals of M_A(G) and those of Mat_n(M_A(G);G) is investigated. It is shown that if every ideal of M_A(G) is an annihilator ideal, then there is a bijection between the ideals of M_A(G) and those of Mat_n(M_A(G);G).1991 Mathematics Subject Classification: 16Y30     

C*-index of observable algebras in <Emphasis Type="Italic">G</Emphasis>-spin model

In two-dimensional lattice spin systems in which the spins take values in a finite group G, one can define a field algebra F which carries an action of a Hopf algebra D(G), the double algebra of G and moreover, an action of D(G;H), which is a subalgebra of D(G) determined by a subgroup H of G, so that F becomes a modular algebra. The concrete construction of D(G;H)-invariant subspace A _H in F is given. By constructing the quasi-basis of conditional expectation γ _G of A _H onto A _G , the C*-index of γ _G is exactly the index of H in G.     

Straightening and bounded cohomology of hyperbolic groups

It was stated by M. Gromov [Gr2] that, for any hyperbolic group G, the map from bounded cohomology Hⁿ_b(G,\Bbb R) H^n_b(G,{\Bbb R}) to Hⁿ(G,\Bbb R) H^n(G,{\Bbb R}) induced by inclusion is surjective for n 3 2 n \ge 2 . We introduce a homological analogue of straightening simplices, which works for any hyperbolic group. This implies that the map Hⁿ_b(G,V) ? Hⁿ(G,V) H^n_b(G,V) \to H^n(G,V) is surjective for n 3 2 n \ge 2 when V is any bounded \Bbb QG {\Bbb Q}G -module and when V is any finitely generated abelian group.     

On the number of colorings of a snark minus an edge

For a given snark G and a given edge e of G, let ψ(G, e) denote the nonnegative integer such that for a cubic graph conformal to G – {e}, the number of Tait colorings with three given colors is 18 · ψ(G, e). If two snarks G₁ and G₂ are combined in certain well‐known simple ways to form a snark G, there are some connections between ψ (G₁, e₁), ψ (G₂, e₂), and ψ(G, e) for appropriate edges e₁, e₂, and e of G₁, G₂, and G. As a consequence, if j and k are each a nonnegative integer, then there exists a snark G with an edge e such that ψ(G, e) = 2^j · 3^k. © 2005 Wiley Periodicals, Inc.     

Sums of Element Orders in Finite Groups

Given a finite group G, write ψ(G) to denote the sum of the orders of the elements of G. Our main result is that if C is a cyclic group and G is a noncyclic group of the same order, then ψ(G) < ψ(C).     

Verifying Huppert’s Conjecture for PSp<Subscript>4</Subscript>(<Emphasis Type="Italic">q</Emphasis>) when <Emphasis Type="Italic">q</Emphasis> > 7

Let G denote a finite group and cd (G) the set of irreducible character degrees of G. Bertram Huppert conjectured that if H is a finite nonabelian simple group such that cd (G) = cd (H), then G ≅ H × A, where A is an abelian group. Huppert verified the conjecture for PSp₄(q) when q = 3, 4, 5, or 7. In this paper, we extend Huppert’s results and verify the conjecture for PSp₄(q) for all q. This demonstrates progress toward the goal of verifying the conjecture for all nonabelian simple groups of Lie type of rank two.     

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.