A base B for a finite permutation group G acting on a set X is a subset of X with the property that only the identity of G can fix every point of B. We prove that a primitive diagonal group G has a base of size 2 unless the top group of G is the alternating or symmetric group acting naturally, in which case a tight bound for the minimal base size of G is given. This bound also satisfies a well-known conjecture of Pyber. Moreover, we prove that if the top group of G does not contain the alternating group, then the proportion of pairs of points that are bases for G tends to 1 as |G| tends to infinity. A similar result for the case when the degree of the top group is fixed is given.Comment: 24 page
Active biological flow networks pervade nature and span a wide range of scales, from arterial blood vessels and bronchial mucus transport in humans to bacterial flow through porous media or plasmodial shuttle streaming in slime molds. Despite their ubiquity, little is known about the self-organization principles that govern flow statistics in such nonequilibrium networks. Here we connect concepts from lattice field theory, graph theory, and transition rate theory to understand how topology controls dynamics in a generic model for actively driven flow on a network. Our combined theoretical and numerical analysis identifies symmetry-based rules that make it possible to classify and predict the selection statistics of complex flow cycles from the network topology. The conceptual framework developed here is applicable to a broad class of biological and nonbiological far-from-equilibrium networks, including actively controlled information flows, and establishes a correspondence between active flow networks and generalized ice-type models.networks | active transport | stochastic dynamics | topology B iological flow networks, such as capillaries (1), leaf veins (2) and slime molds (3), use an evolved topology or active remodeling to achieve near-optimal transport when diffusion is ineffectual or inappropriate (2, 4-7). Even in the absence of explicit matter flux, living systems often involve flow of information currents along physical or virtual links between interacting nodes, as in neural networks (8), biochemical interactions (9), epidemics (10), and traffic flow (11). The ability to vary the flow topology gives networkbased dynamics a rich phenomenology distinct from that of equivalent continuum models (12). Identical local rules can invoke dramatically different global dynamical behaviors when node connectivities change from nearest-neighbor interactions to the broad distributions seen in many networks (13-16). Certain classes of interacting networks are now sufficiently well understood to be able to exploit their topology for the control of input-output relations (17, 18), as exemplified by microfluidic logic gates (19,20). However, when matter or information flow through a noisy network is not merely passive but actively driven by nonequilibrium constituents (3), as in maze-solving slime molds (6), there are no overarching dynamical self-organization principles known. In such an active network, noise and flow may conspire to produce behavior radically different from that of a classical forced network. This raises the general question of how path selection and flow statistics in an active flow network depend on its interaction topology.Flow networks can be viewed as approximations of a complex physical environment, using nodes and links to model intricate geometric constraints (21-23). These constraints can profoundly affect matter transport (24-27), particularly for active systems (28-30) where geometric confinement can enforce highly ordered collective dynamics (20,(31)(32)(33)(34)(35)(36)(37)(38)(39)(40). In s...
Given a finite group G and a faithful irreducible F G-module V where F has prime order, does G have a regular orbit on V ? This problem is equivalent to determining which primitive permutation groups of affine type have a base of size 2. In this paper, we classify the pairs (G, V ) for which G has a regular orbit on V where G is a covering group of a symmetric or alternating group and V is a faithful irreducible F G-module such that the order of F is prime and divides the order of G.
Given a finite group G and a faithful irreducible F G-module V where F has prime order, does G have a regular orbit on V ? This problem is equivalent to determining which primitive permutation groups of affine type have a base of size 2. Let G be a covering group of an almost simple group whose socle T is sporadic, and let V be a faithful irreducible F G-module where F has prime order dividing |G|. We classify the pairs (G, V ) for which G has no regular orbit on V , and determine the minimal base size of G in its action on V . To obtain this classification, for each non-trivial g ∈ G/Z(G), we compute the minimal number of T -conjugates of g generating T, g .If there are exactly m faithful irreducible F p G-modules with dimension d on which G has no regular orbit and m > 1, then we write d (m) in Table 1. Except for the case (G, p, dim Fp (V )) = (M 11 , 3, 10), this is sufficient to identify the non-regular modules. However, there are three faithful irreducible F 3 M 11 -modules of dimension 10, only one of which Key words and phrases. regular orbit; base size; sporadic simple group; primitive affine group.
Abstract. Locally triangular graphs are known to be halved graphs of bipartite rectagraphs, which are connected triangle-free graphs in which every 2-arc lies in a unique quadrangle. A graph Γ is locally rank 3 if there exists G Aut(Γ) such that for each vertex u, the permutation group induced by the vertex stabiliser Gu on the neighbourhood Γ(u) is transitive of rank 3. One natural place to seek locally rank 3 graphs is among the locally triangular graphs, where every induced neighbourhood graph is isomorphic to a triangular graph Tn. This is because the graph Tn, which has vertex set the 2-subsets of {1, . . . , n} and edge set the pairs of 2-subsets intersecting at one point, admits a rank 3 group of automorphisms. In this paper, we classify the locally 4-homogeneous rectagraphs under some additional structural assumptions. We then use this result to classify the connected locally triangular graphs that are also locally rank 3.
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