120974879
Presentations for (G,s)-transitive graphs of small valency
Given a group G , a subgroup H and an element a ∈ G , we define Γ( G, H, a ) to be the graph on the set of left-cosets of H in G , where two left-cosets g 1 H and g 2 H are adjacent whenever . We will consider this construction only in the case that The first of these conditions assures that adjacency is a symmetric relation (i.e. that Γ( G, H, a ) is an undirected graph) and the second assures that Γ( G, H, a ) is connected.
120339122
Extended generalized hexagons
119635932
On Distance-Transitive Graphs
On suppose que Γ est un graphe non oriente connexe de genre fini g et G est un sous-groupe de aut(Γ) tel que pour chaque sommet x, G(x) est fini et agit transitivement sur l'ensemble des s-chemins (x 0 ,…,x s ) avec x 0 =x. Soit M un entier positif avec M≡3 (mod 8) et soit m=(M+9)/4 et n=max(2m+4,14). Si s≥2(m−1) et s≥(g−M)/2 alors Gm(x 0 ,…,x n )=1 pour chaque n-chemin (x 0 ,…,x n )
46019521
The nonexistence of 8-transitive graphs
We prove that the inequalitys≦7 holds for finites-transitive graphs assuming that the list of known 2-transitive permutation groups is complete.
121605215
On finite Moufang polygons
We show that a finite generalized polygon Γ is Moufang with respect to a groupG if and only if for every flag {x, y} of Γ, the subgroupG1(x, y) ofG fixing every element incident with one ofx, y acts transitively on the set of apartments containing the elementsu, x, y, w, whereu≠y (resp.w≠x) is an arbitrary element incident withx (resp.y).
122472158
3-transpositions in infinite groups
Let G be a group. A subset D will be called a set of 3-transpositions if | x | = 2 for x e D and | xy | = 3 whenever x, yeD do not commute. We will call the set D closed if xDx −1 = D for each xeD . For each xeD , let For each subset X of D , we denote by [ X ] the graph with vertex set X where two elements x, yeX are joined by an edge whenever they commute. We denote by ( X ) the complement graph; thus two elements x, yeX are joined by an edge of ( X ) whenever they do not commute.
19510229
Bifurcation in difference approximations to two-point boundary value problems
Numerical methods for bifurcation problems of the form (*) LyAXf(y), By =0, where f(0) = 0 and f'(0) * 0, are considered. Here y is a scalar function, X is a real scalar, L is a linear differential operator and By = 0 represents some linear homogeneous two-point boundary conditions. Under certain assumptions, it is shown that if (*) is replaced by an appropriate difference scheme, then there exists a unique branch of nontrivial solutions of the discrete problem in a neighborhood of a branch of nontrivial solutions of (*) bifurcating from the trivial solution and that the discrete branch converges to the continuous one. Error estimates are derived and an illustrative numerical example is included.
120998826
Symmetric graphs with projective subconstituents
Let P be an undirected graph and G a subgroup of aut(r) acting transitively on the vertex set of P. Let x be an arbitrary vertex of P. We denote by P(x) the set of vertices adjacent to x and by G(x)F(x) the permutation group induced by the stabilizer G (x) of x in G on P(x); G (x)r(X) is called the subconstituent of the pair (P, G). For each i E N let Gi(x) = {a E G(x)Ia E G(y) if a (x, y) 1 then s 3. In [8], it was shown that s 3. We prove the following partial result:
122289259
On the Approximation of Fixed Points of Nonlinear Compact Operators
Certain methods for the approximate solution of $x - Kx = 0$, where K is a compact operator, lead to a family of operator equations $x - K_n x = 0$, $n \geqq 1$, where $K_n $,$n \geqq 1$, are collectively compact and converge pointwise to K. In this paper a new and simplified proof of the existence and uniqueness of a solution of $x - K_n x = 0$ in a neighborhood of an isolated solution of $x - Kx = 0$ is given.
6645734
A Geometric Characterization of Certain Groups of Lie Type
Let r be an undirected graph with vertex set V(r) and edge set E(r) and let G be a subgroup of Aut (r). For each {x, y} E E(r), we denote by G{x, y} the stabilizer of {x, y} in G (which mayor may not contain elements exchanging the vertices x and y). For each x E V(r) we denote by r(x) the set of vertices adjacent to x, by G(x) the stabilizer of x in G and, for each i E N, by Gi(x) the subgroup {a E Gla E G(u) for all u E V(r) with a(x, u) "'" i} where a(x, u) denotes the distance from x to u. An s-path (for sEN) is an (s + I)-tuple (xo, Xl. .•• , xs ) of vertices such that Xi E r(Xi-l) if 1"", i "'" s and Xi ¥Xi-Z if 2"", i "'" s. Let G(xo, ... , xs ) = G(xo) n··· n G(xs ) and Gi(xo, ... , xs ) = Gi(xo) n· .. n Gi(xs ) for each s-path (xo, ... , xs ) and each i E N. If H is any group acting on a set X, we denote by H X the permutation group on X induced by H. . In [12] we proved the following result.
53988483
Permutation groups with projective unitary subconstituents
AssTRAcT. Let r be a finite directed graph with vertex set V(I) and edge set EQ7) and let G be a subgroup of aut(I) which we assume to act transitively on both V(r) and E(r). Suppose that for some prime power q, the stabilizer G(x) of a vertex x induces on both {yI(x,y) E E(r)) and {wl(w, x) E E)} a group lying between PSU(3, q2) and Pr1U(3, q2). It is shown that if G acts primitively on V(I), then for each edge (x, y), the subgroup of G(x) fixing every vertex in {wI(x, w) or (y, w) E E(r)) is trivial.