Finite groups with three conjugacy class sizes of some elements

Let G be a finite group. We prove as follows: Let G be a p-solvable group for a fixed prime p. If the conjugacy class sizes of all elements of primary and biprimary orders of G are {1,p ^a , n} with a and n two positive integers and (p,n)?=?1, then G is p-nilpotent or G has abelian Sylow p-subgroups.     

An extension of a theorem of Alan Camina on conjugacy class sizes

Let G be a finite group. Let n be a positive integer and p a prime coprime to n. In this paper we prove that if the set of conjugacy class sizes of primary and biprimary elements of group G is {1,p ^a , p ^a n}, then G ≌ G ₀ × H, where H is abelian and G ₀ contains a normal subgroup M × P ₀ of index p. Moreover, M × P ₀ is the set of all elements of G ₀ of conjugacy class sizes p ^a or 1, where M is an abelian π(n)-subgroup of G ₀ and P ₀ is an abelian p-subgroup of G ₀, neither being central in G. Finally, p ^a = p and P/P ₀ acts fixed-point-freely on M and ?(P) ≤ Z(P). This is an extension of Alan Camina’s theorems on the structure of groups whose set of conjugacy class size is {1,p ^a , p ^a q ^b }, where p and q are two distinct primes.     

On the spanning connectivity of graphs

A k-containerC(u,v) of G between u and v is a set of k internally disjoint paths between u and v. A k-container C(u,v) of G is a k^*-container if it contains all vertices of G. A graph G is k^*-connected if there exists a k^*-container between any two distinct vertices. The spanning connectivity of G, κ^*(G), is defined to be the largest integer k such that G is w^*-connected for all 1?w?k if G is a 1^*-connected graph. In this paper, we prove that κ^*(G)?2δ(G)-n(G)+2 if (n(G)/2)+1?δ(G)?n(G)-2. Furthermore, we prove that κ^*(G-T)?2δ(G)-n(G)+2-|T| if T is a vertex subset with |T|?2δ(G)-n(G)-1.     

On an extension of a theorem on conjugacy class sizes

Let G be a finite group. We extend Alan Camina’s theorem on conjugacy classes sizes which asserts that if the conjugacy classes sizes of G are {1, p ^a , q ^b , p ^a q ^b }, where p and q are two distinct primes and a and b are integers, then G is nilpotent. We show that let G be a group and assume that the conjugacy classes sizes of elements of primary and biprimary orders of G are exactly {1, p ^a , n,p ^a n} with (p, n) = 1, where p is a prime and a and n are positive integers. If there is a p-element in G whose index is precisely p ^a , then G is nilpotent and n = q ^b for some prime q ≠ p.     

A solution concept forn-person noncooperative games

The paper describes a solution concept forn-person noncooperative games, developed jointly by the author and Reinhard Selten. Its purpose is to select one specific perfect equilibrium points=s (G) as the solution of any given noncooperative gameG. The solution is constructed by an inductive procedure. In defining the solutions (G) of gameG, we use the solutionss (G ^*) of the component gamesG ^* (if any) ofG; and in defining the solutions (G^*) of any such component gameG ^*, we use the solutionss (G ^**) of its own component gamesG ^** (if any), etc. This inductive procedure is well-defined because it always comes to an end after a finite number of steps. At each level, the solution of a game (or of a component game) is defined in two steps. First, aprior subjectiveprobability distribution p _i is assigned to the pure strategies of each playeri, meant to represent the other players' initial expectations about playeri's likely strategy choice. Then, a mathematical procedure, called thetracing procedure, is used to define the solution on the basis of these prior probability distributionsp _i . The tracing procedure is meant to provide a mathematical representation for thesolution process by which rational players manage to coordinate their strategy plans and their expectations, and make them converge to one specific equilibrium point as solution for the game     

A note on acyclic edge coloring of complete bipartite graphs

An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic (2-colored) cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and is denoted by a^′(G). Let Δ=Δ(G) denote the maximum degree of a vertex in a graph G. A complete bipartite graph with n vertices on each side is denoted by K_n,n. Alon, McDiarmid and Reed observed that a^′(K_p−1,p−1)=p for every prime p. In this paper we prove that a^′(K_p,p)≤p+2=Δ+2 when p is prime. Basavaraju, Chandran and Kummini proved that a^′(K_n,n)≥n+2=Δ+2 when n is odd, which combined with our result implies that a^′(K_p,p)=p+2=Δ+2 when p is an odd prime. Moreover we show that if we remove any edge from K_p,p, the resulting graph is acyclically Δ+1=p+1-edge-colorable.     

Explicit constants for Hardy’s inequality with power weight on n-dimensional product spaces

In this paper, Hardy operator H on n-dimensional product spaces G = (0, ∞)n and its adjoint operator H* are investigated. We use novel methods to obtain two main results. One is that we characterize the sufficient and necessary conditions for the operators H and H* being bounded from Lp(G, xα) to Lq(G, xβ), and the bounds of the operators H and H* are explicitly worked out. The other is that when 1 < p = q < +∞, norms of the operators H and H* are obtained.     

The Proof of a Conjecture of Additive Number Theory

The aim of this paper is to show that for any n∈N, n>3, there exist a, b∈N* such that n=a+b, the “lengths” of a and b having the same parity (see the text for the definition of the “length” of a natural number). Also we will show that for any n∈N, n>2, n≠5, 10, there exist a, b∈N* such that n=a+b, the “lengths” of a and b having different parities. We will prove also that for any prime p≡7(mod 8) there exist a, b∈N* such that p=a²+b, the “length” of b being an even number.     

Linear equations with unknowns from a multiplicative group whose solutions lie in a small number of subspaces

Let K be a field of characteristic 0 and let (K^*)ⁿ denote the n-fold Cartesian product of K^*, endowed with coordinatewise multiplication. Let Γ be a subgroup of (K^*)ⁿ of finite rank. We consider equations (*) a₁x₁ + … + a_nx_n = 1 in x = (x₁x_n)Γ, where a = (a₁,a_n)(K^*)ⁿ. Two tuples a, b(K^*)ⁿ are called Γ-equivalent if there is a uΓ such that b = u · a. Gy?ry and the author [Compositio Math. 66 (1988) 329-354] showed that for all but finitely many Γ-equivalence classes of tuples a(K^*)ⁿ, the set of solutions of (*) is contained in the union of not more than 2⁽ⁿ⁺¹! proper linear subspaces of Kⁿ. Later, this was improved by the author [J. reine angew. Math. 432 (1992) 177-217] to (n!)²ⁿ⁺². In the present paper we will show that for all but finitely many Γ-equivalence classes of tuples of coefficients, the set of non-degenerate solutions of (*) (i.e., with non-vanishing subsums) is contained in the union of not more than 2ⁿ proper linear subspaces of Kⁿ. Further we give an example showing that 2ⁿ cannot be replaced by a quantity smaller than n.     

Sur le changement de base stable des intégrales orbitales pondérées

Let G^′ be a quasi-split connected reductive group over a p-adic field F. Let E be a cyclic extension of F. In the context of cyclic base change, we can attach to G^′ and E a twisted space G^* (in the sense of Labesse). Let G be an inner form of G^*. If G^′ is GL(n), SL(n) or more generally a group which we call L-stable, we define and prove the existence of a non-invariant transfer between the weighted orbital integrals of G and those of G^′. For GL(n), such a transfer has been conjectured by Labesse. The proof is based on previous results of harmonic analysis on Lie algebras and on a generalization of a result of Waldspurger concerning Arthur's (G,M)-families.     

On the existence of super P-groups

A finite group (G, ·) is said to be sequenceable if its elements can be arranged in a sequence a₀ = e, a₁, a₂,…, a_n?1 in such a way that the partial products b₀ = a₀, b₁ = a₀a₁, b₂ = a₀a₁a₂,…, b_n?1 = a₀a₁a₂ ··· a_n?1 are all distinct (and consequently are the elements of G in a new order). It is said to be R-sequenceable if its elements can be ordered in such a way that the partial products b₀ = a₀, b₁ = a₀a₁, b₂ = a₀a₁a₂,…, b_n?2 = a₀a₁a₂ ··· a_n?2 are all different and so that b_n?1 = a₀a₁a₂ ··· a_n?1 = b₀ = e. (in the first case, the ordering a₀,a₁,…,a_n?1 of the elements is said to be a sequencing of G and, in the second case, an R-sequencing of G.) It is a super P-group if every element of one particular coset hG′ of the derived group can be expressed as the product of the n elements of G in such a way that the orderings of the elements in these products are sequencings of G with the exception that, in the case that h = e, the element e of G′ must be expressed as a product of the n elements of G which forms an R-sequencing of G. It is proved that every non-Abelian group of order pq such that p has 2 as a primitive root, where p and q are distinct odd primes with p < q, is a super P-group. Also provided is evidence for the conjectures that all Abelian groups and all dihedral groups of doubly even order (except those of orders 4 and 8) are super P-groups.     

The continuum limit of critical random graphs

We consider the Erd?s–Rényi random graph G(n, p) inside the critical window, that is when p?=?1/n?+?λn ^?4/3, for some fixed ${\lambda \in \mathbb{R}}$ . We prove that the sequence of connected components of G(n, p), considered as metric spaces using the graph distance rescaled by n ^?1/3, converges towards a sequence of continuous compact metric spaces. The result relies on a bijection between graphs and certain marked random walks, and the theory of continuum random trees. Our result gives access to the answers to a great many questions about distances in critical random graphs. In particular, we deduce that the diameter of G(n, p) rescaled by n ^?1/3 converges in distribution to an absolutely continuous random variable with finite mean.     

Matroids over fp which are rational excluded minors

For a given prime p, we construct a collection of 2^p matroids G_p,a with (1) χ_pf(G_p,a)={p}, and (2) G_p,a is an excluded minor for rational representability. The motivating construction (Section 2) disproves a conjectures of Reid [4], using relatively high-rank, high cardinality matroids. The general construction (Section 3) makes use of ordered partitions (χ_pf(G) denotes the prime-field characteristic set of G, i.e., the set of prime fields over which G may be represented, while G can be represented over fields of no other characteristic.) Finally, Section 4 offers another construction with the same properties–a kind of projective dual to Section 2.     

Groups with a positive law on commutators

Let p be a prime number and let G be a finitely generated group that is residually a finite p-group. We prove that if G satisfies a positive law on all elements of the form [a,b][c,d]ⁱ, a,b,c,d∈G and i?0, then the entire derived subgroup G^′ satisfies a positive law. In fact, G^′ is an extension of a nilpotent group by a locally finite group of finite exponent.     

Optimal covers with Hamilton cycles in random graphs

A packing of a graph G with Hamilton cycles is a set of edge-disjoint Hamilton cycles in G. Such packings have been studied intensively and recent results imply that a largest packing of Hamilton cycles in G _n,p a.a.s. has size ?δ(G _n,p )/2?. Glebov, Krivelevich and Szabó recently initiated research on the ‘dual’ problem, where one asks for a set of Hamilton cycles covering all edges of G. Our main result states that for \(\tfrac{{log^{117} n}} {n} \leqslant p \leqslant 1 - n^{ - 1/8}\) , a.a.s. the edges of G _n,p can be covered by ?Δ (G _n,p )/2? Hamilton cycles. This is clearly optimal and improves an approximate result of Glebov, Krivelevich and Szabó, which holds for p ≥ n ^?1+?. Our proof is based on a result of Knox, Kühn and Osthus on packing Hamilton cycles in pseudorandom graphs.     

On the primes withp n+1 -p n = 8 and the Sum of their Reciprocals

We introduce the counting function π _2,8 ^* (x) of the primes with difference 8 between consecutive primes ( ****p _n,p_n+1 =p _n + 8) can be approximated by logarithm integralLi _2,8 ^* . We calculate the values of π _2,8 ^* (x) and the sumC _2,8(x) of reciprocals of primes with difference 8 between consecutive primes (p _n,p_n+1 =p _n +8)) wherex is counted up to 7 x 10¹⁰. From the results of these calculations, we obtain π _2,8 ^* (7 x 10¹⁰) = 133295081 andC _2,8(7 x 10¹⁰) = 0.3374 ±2.6 x 10^-4.     

On dimensional rigidity of bar-and-joint frameworks

Let V={1,2,…,n}. A mapping p:V→R^r, where p¹,…,pⁿ are not contained in a proper hyper-plane is called an r-configuration. Let G=(V,E) be a simple connected graph on n vertices. Then an r-configuration p together with graph G, where adjacent vertices of G are constrained to stay the same distance apart, is called a bar-and-joint framework (or a framework) in R^r, and is denoted by G(p). In this paper we introduce the notion of dimensional rigidity of frameworks, and we study the problem of determining whether or not a given G(p) is dimensionally rigid. A given framework G(p) in R^r is said to be dimensionally rigid iff there does not exist a framework G(q) in R^s for s?r+1, such that ∥qⁱ-q^j∥²=∥pⁱ-p^j∥² for all (i,j)∈E. We present necessary and sufficient conditions for G(p) to be dimensionally rigid, and we formulate the problem of checking the validity of these conditions as a semidefinite programming (SDP) problem. The case where the points p¹,…,pⁿ of the given r-configuration are in general position, is also investigated.     

The fitting length of solvableH pn-groups

For a (finite) groupG and some prime powerp ⁿ, theH _p ⁿ -subgroupH _pn (G) is defined byH _p ⁿ (G)=〈xεG|x ^pn≠1〉. A groupH≠1 is called aH _p ⁿ -group, if there is a finite groupG such thatH is isomorphic toH _p ⁿ (G) andH _p ⁿ (G)≠G. It is known that the Fitting length of a solvableH _p ⁿ -group cannot be arbitrarily large: Hartley and Rae proved in 1973 that it is bounded by some quadratic function ofn. In the following paper, we show that it is even bounded by some linear function ofn. In view of known examples of solvableH _p ⁿ -groups having Fitting lengthn, this result is “almost” best possible.     

Component structure of the vacant set induced by a random walk on a random graph

We consider random walks on several classes of graphs and explore the likely structure of the vacant set, i.e. the set of unvisited vertices. Let Γ(t) be the subgraph induced by the vacant set of the walk at step t. We show that for random graphs G_n,p (above the connectivity threshold) and for random regular graphs G_r,r ≥ 3, the graph Γ(t) undergoes a phase transition in the sense of the well‐known ErdJW‐RSAT1100590x.png ‐Renyi phase transition. Thus for t ≤ (1 ‐ ε)t^*, there is a unique giant component, plus components of size O(log n), and for t ≥ (1 + ε)t^* all components are of size O(log n). For G_n,p and G_r we give the value of t^*, and the size of Γ(t). For G_r, we also give the degree sequence of Γ(t), the size of the giant component (if any) of Γ(t) and the number of tree components of Γ(t) of a given size k = O(log n). We also show that for random digraphs D_n,p above the strong connectivity threshold, there is a similar directed phase transition. Thus for t ≤ (1 ‐ ε)t^*, there is a unique strongly connected giant component, plus strongly connected components of size O(log n), and for t ≥ (1 + ε)t^* all strongly connected components are of size O(log n). © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2012