Involutory Automorphisms of Groups of Odd Order

The following theorem is proved. Let G be a finite group of odd order admitting an involutory automorphism φ. Suppose that G has derived length d and that C_G(φ) is nilpotent of class c. Assume that C_G(φ) is a m-generator. Then [G,φ]^′ is nilpotent of {c,d,m}-bounded class.     

Centralizers of coprime automorphisms of finite groups

Let A be an elementary abelian group of order p ^k with k ≥ 3 acting on a finite p′-group G. The following results are proved. If γ _k-2(C _G (a)) is nilpotent of class at most c for any ${a \in A^{\#}}$ , then γ _k-2(G) is nilpotent and has {c, k, p}-bounded nilpotency class. If, for some integer d such that 2 ^d  + 2 ≤ k, the dth derived group of C _G (a) is nilpotent of class at most c for any ${a \in A^{\#}}$ , then the dth derived group G ^(d) is nilpotent and has {c, k, p}-bounded nilpotency class.     

P. Hall's covering group and embeddings of countable cc-groups

Let A be an elementary abelian group of order at least p ³ acting on a finite p′-group G that is soluble with derived length d. Assume that γ _c (C _G (a)) has exponent dividing m for any a ∈ A ^#. It is proved that there exist {p, d, c, m}-bounded numbers c ₁ and m ₁ such that γ _{c 1} (G) has exponent dividing m ₁.     

Properly colored subgraphs and rainbow subgraphs in edge‐colorings with local constraints

We consider a canonical Ramsey type problem. An edge‐coloring of a graph is called m‐good if each color appears at most m times at each vertex. Fixing a graph G and a positive integer m, let f(m, G) denote the smallest n such that every m‐good edge‐coloring of K_n yields a properly edge‐colored copy of G, and let g(m, G) denote the smallest n such that every m‐good edge‐coloring of K_n yields a rainbow copy of G. We give bounds on f(m, G) and g(m, G). For complete graphs G = K_t, we have c₁mt²/ln t ≤ f(m, K_t) ≤ c₂mt², and cmt³/ln t ≤ g(m, K_t) ≤ cmt³/ln t, where c₁, c₂, c, c are absolute constants. We also give bounds on f(m, G) and g(m, G) for general graphs G in terms of degrees in G. In particular, we show that for fixed m and d, and all sufficiently large n compared to m and d, f(m, G) = n for all graphs G with n vertices and maximum degree at most d. © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 2003     

The vertex-cover polynomial of a graph

In this paper we define the vertex-cover polynomial Ψ(G,τ) for a graph G. The coefficient of τ^r in this polynomial is the number of vertex covers V′ of G with |V′|=r. We develop a method to calculate Ψ(G,τ). Motivated by a problem in biological systematics, we also consider the mappings f from {1, 2,…,m} into the vertex set V(G) of a graph G, subject to f⁻¹(x)f⁻¹(y)≠ for every edge xy in G. Let F(G,m) be the number of such mappings f. We show that F(G,m) can be determined from Ψ(G,τ).     

Bounded edge-connectivity and edge-persistence of Cartesian product of graphs

The bounded edge-connectivity λ_k(G) of a connected graph G with respect to is the minimum number of edges in G whose deletion from G results in a subgraph with diameter larger than k and the edge-persistence D⁺(G) is defined as λ_d(G)(G), where d(G) is the diameter of G. This paper considers the Cartesian product G₁×G₂, shows λ_k1+k2(G₁×G₂)≥λ_k1(G₁)+λ_k2(G₂) for k₁≥2 and k₂≥2, and determines the exact values of D⁺(G) for G=C_n×P_m, C_n×C_m, Q_n×P_m and Q_n×C_m.     

Pairwise intersections and forbidden configurations

Let f_m(a,b,c,d) denote the maximum size of a family of subsets of an m-element set for which there is no pair of subsets with

By symmetry we can assume a≥d and b≥c. We show that f_m(a,b,c,d) is Θ(m^a+b−1) if either b>c or a,b≥1. We also show that f_m(0,b,b,0) is Θ(m^b) and f_m(a,0,0,d) is Θ(m^a). The asymptotic results are as m→∞ for fixed non-negative integers a,b,c,d. This can be viewed as a result concerning forbidden configurations and is further evidence for a conjecture of Anstee and Sali. Our key tool is a strong stability version of the Complete Intersection Theorem of Ahlswede and Khachatrian, which is of independent interest.     

Majorants and Minorants for Elliptic Measures on

Let μ(· ; Σ, G₁) and μ(· ; Ω, G₂) be elliptically contoured measures on ^k centered at 0, having scale parameters (Σ, Ω) and radial cdf′s (G₁, G₂). Elliptical measures v_m(·) and v_M(·), depending on (Σ, Ω, G₁, G₂), are constructed such that V_m(C) ≤ {μ(C; Σ, G₁), μ(C; Ω, G₂)} for every symmetric convex set C ^k with equality for certain sets. These in turn rely on the construction of spectral lower and upper matrix bounds for (Σ, Ω). Extensions include bounds for certain ensembles and mixtures, including versions having star-shaped contours. The lindings specialize to give envelopes for some nonstandard distributions of quadratic forms, with applications to stochastic characteristics of ballistic systems.     

Reflection sieve obstructions in finite groups of type dn

In [4] we constructed certain homology representations of a finite group G of type A_n, B_n or C_n, and showed that these representations can be used to sift out the reflection compound characters of G. In the present note, we show that for a group G of type D_n, each reflection compound character π^(k), 2 k n − 2, determines a unique “obstruction” character θ^(k), which occurs with positive multiplicity in every homology representation containing π^(k).     

On the relative character correspondences

Let π(G,A):Irr_A(G)→Irr(C_G(A)) be the Glauberman–Isaacs correspondence, where G and A are finite groups with coprime orders and A acts on G by automorphisms. Let B be a subgroup of A. In this setting, we give some new conditions for the fixed-point subgroups C_G(A) and C_G(B) such that χπ(G,A) is an irreducible constituent of the restriction of χπ(G,B) to C_G(A) for all χIrr_A(G).     

On finding the core of a tree with a specified length

A core of a graph G is a path P in G that is central with respect to the property of minizining d(P) = Σ_υεV(G)d(υ, P), where d(υ, P) is the distance from vertex υ to path P. This paper explores some properties of a core of a specified length.     

Large centralizers in finite solvable groups

In 1955 R. Brauer and K. A. Fowler showed that ifG is a group of even order >2, and the order |Z(G)| of the center ofG is odd, then there exists a strongly real) elementx∈G−Z whose centralizer satisfies|C _G(x)|>|G|^1/3. In Theorem 1 we show that every non-abeliansolvable groupG contains an elementx∈G−Z such that|C _G(x)|>[G:G′∩Z]^1/2 (and thus|C _G(x)|>|G|^1/3). We also note that if non-abelianG is either metabelian, nilpotent or (more generally) supersolvable, or anA-group, or any Frobenius group, then|C _G(x)|>|G|^1/2 for somex∈G−Z. In Theorem 2 we prove that every non-abelian groupG of orderp ^mqⁿ (p, q primes) contains a proper centralizer of order >|G|^1/2. Finally, in Theorem 3 we show that theaverage |C(x)|, x∈G, is ≧c|G| ^1/3 for metabelian groups, wherec is constant and the exponent 1/3 is best possible.     

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.     

Spectral Density for Third Order Cumulants under Strong Mixing Conditions

If X{X_v: v ^d} is a strictly stationary random field, with X₀ bounded and expressible as a sum of indicator functions satisfying certain conditions, if the mixing coefficient α(s) is summable over ^d (that, is, ∑_m m^d−1α(m)<∞), and if a mixing condition involving three sets is satisfied, then the third order cumulant Cum(X_a, X_b, X_c) of X has a continuous spectral density. We do not begin with the assumption that the cumulants are absolutely summable.     

Relatively Free Nilpotent Torsion-Free Groups and Their Lie Algebras

Let K be a field of characteristic zero. For a torsion-free finitely generated nilpotent group G, we naturally associate four finite dimensional nilpotent Lie algebras over K, ? _K (G), grad^(?)(? _K (G)), grad^(g)(exp ? _K (G)), and L _K (G). Let 𝔗 _c be a torsion-free variety of nilpotent groups of class at most c. For a positive integer n, with n ≥ 2, let F _n (𝔗 _c ) be the relatively free group of rank n in 𝔗 _c . We prove that ? _K (F _n (𝔗 _c )) is relatively free in some variety of nilpotent Lie algebras, and ? _K (F _n (𝔗 _c )) ? L _K (F _n (𝔗 _c )) ? grad^(?)(? _K (F _n (𝔗 _c ))) ? grad^(g)(exp ? _K (F _n (𝔗 _c ))) as Lie algebras in a natural way. Furthermore, F _n (𝔗 _c ) is a Magnus nilpotent group. Let G ₁ and G ₂ be torsion-free finitely generated nilpotent groups which are quasi-isometric. We prove that if G ₁ and G ₂ are relatively free of finite rank, then they are isomorphic. Let L be a relatively free nilpotent Lie algebra over ? of finite rank freely generated by a set X. Give on L the structure of a group R, say, by means of the Baker–Campbell–Hausdorff formula, and let H be the subgroup of R generated by the set X. We show that H is relatively free in some variety of nilpotent groups; freely generated by the set X, H is Magnus and L ? ?_?(H) ? L _?(H) as Lie algebras. For relatively free residually torsion-free nilpotent groups, we prove that ? _K and L _K are isomorphic as Lie algebras. We also give an example of a finitely generated Magnus nilpotent group G, not relatively free, such that ?_?(G) is not isomorphic to L _?(G) as Lie algebras.     

Longest Paths and Longest Cycles in Graphs with Large Degree Sums

 Let p(G) and c(G) denote the number of vertices in a longest path and a longest cycle, respectively, of a finite, simple graph G. Define σ₄(G)=min{d(x ₁)+d(x ₂)+ d(x ₃)+d(x ₄) | {x ₁,…,x ₄} is independent in G}. In this paper, the difference p(G)−c(G) is considered for 2-connected graphs G with σ₄(G)≥|V(G)|+3. Among others, we show that p(G)−c(G)≤2 or every longest path in G is a dominating path. Received: August 28, 2000 Final version received: May 23, 2002     

Prime ideals in the C*-algebra of a nilpotent group

We say that a locally compact groupG hasT ₁ primitive ideal space if the groupC ^*-algebra,C ^*(G), has the property that every primitive ideal (i.e. kernel of an irreducible representation) is closed in the hull-kernel topology on the space of primitive ideals ofC ^*(G), denoted by PrimG. This means of course that every primitive ideal inC ^*(G) is maximal. Long agoDixmier proved that every connected nilpotent Lie group hasT ₁ primitive ideal space. More recentlyPoguntke showed that discrete nilpotent groups haveT ₁ primitive ideal space and a few month agoCarey andMoran proved the same property for second countable locally compact groups having a compactly generated open normal subgroup. In this note we combine the methods used in [3] with some ideas in [9] and show that for nilpotent locally compact groupsG, having a compactly generated open normal subgroup, closed prime ideals inC ^*(G) are always maximal which implies of course that PrimG isT ₁.     

Nilpotency of finite groups with Frobenius groups of automorphisms

Suppose that a finite group G admits a Frobenius group of automorphisms BC of coprime order with kernel B and complement C such that C _G (C) is abelian. It is proved that if B is abelian of rank at least two and [C_G(u), C_G(v),...,C_G(v)]=1{[C_G(u), C_G(v),\dots,C_G(v)]=1} for any u,v ? B\{1}{u,v\in B{\setminus}\{1\}}, where C _G (v) is repeated k times, then G is nilpotent of class bounded in terms of k and |C| only. It is also proved that if B is abelian of rank at least three and C _G (b) is nilpotent of class at most c for every b ? B\{1}{b \in B{\setminus}\{1\}}, then G is nilpotent of class bounded in terms of c and |C|. The proofs are based on results on graded Lie rings with many commuting components.     

Pointwise bounds on eigenfunctions and wave packets in N-body quantum systems. VI. Asymptotics in the two-cluster region

We study the asymptotic behavior of the ground-state wave function of multiparticle quantum systems without statistics in that region of configuration space where the particles break up into two well-defined clusters very far apart. One example of our results is the following: consider a system of N particles moving in three dimensions with rotationally invariant two-body potentials which are bounded and have compact support. Let D = C₁,C₂ be a partition into two clusters so that H(C₁) and H(C₂) have discrete ground states η₁ and η₂ of energy ε₁ and ε₂. Suppose that Σ = ε₁ + ε₂ = inf σ_ess(H) and that H has a discrete ground state of energy E. Let ζ₁and ζ₂ denote internal coordinates for the clusters C₁ and c₂ and let R be the difference of the centers of mass of the clusters. Let μ = M₁M₂/M₁ + M₂with M_i the mass of clusters C_i and define k by k²/2m = Σ-E. Then as R → a8 with ¦ζ_i¦ bounded, we prove that (ζ₁,ζ₂, R) = cη(ζ₁)η(ζ₂)e^−kRR⁻¹(1+O(e^−γR)) for some γ, c > 0. We prove weaker conclusions under weaker hypotheses, including results in the atomic case.     

The Ramsey numbers for cycles versus wheels of even order

For two given graphs G₁ and G₂, the Ramsey number R(G₁,G₂) is the smallest integer n such that for any graph G of order n, either G contains G₁ or the complement of G contains G₂. Let C_n denote a cycle of order n and W_m a wheel of order m+1. Surahmat, Baskoro and Tomescu conjectured that R(C_n,W_m)=3n−2 for m odd, n≥m≥3 and (n,m)≠(3,3). In this paper, we confirm the conjecture for n≥20.