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A group is 3 2 -generated if every non-identity element is contained in a generating pair. A conjecture of Breuer, Guralnick and Kantor from 2008 asserts that a finite group is 3 2 -generated if and only if every proper quotient of the group is cyclic, and recent work of Guralnick reduces this conjecture to almost simple groups. In this paper, we prove a stronger form of the conjecture for almost simple symplectic and odd-dimensional orthogonal groups. More generally, we study the uniform spread of these groups, obtaining lower bounds and related asymptotics. This builds on earlier work of Burness and Guest, who established the conjecture for almost simple linear groups. [ABSTRACT FROM AUTHOR]

Abstract: We consider two variants of the task of spreading a swarm of agents uniformly on a ring graph. Ant-like oblivious agents having limited capabilities are considered. The agents are assumed to have little memory, they all execute the same algorithm and no direct communication is allowed between them. Furthermore, the agents do not possess any global information. In particular, the size of the ring () and the number of agents in the swarm () are unknown to them. The agents are assumed to operate on an unweighted ring graph. Every agent can measure the distance to his two neighbors on the ring, up to a limited range of edges. The first task considered, is dynamical (i.e. in motion) uniform deployment on the ring. We show that if either the ring is unoriented, or the visibility range is less than , this is an impossible mission for the agents. Then, for an oriented ring and , we propose an algorithm which achieves the deployment task in optimal time. The second task discussed, called quiescent spread, requires the agents to spread uniformly over the ring and stop moving. We prove that under our model, in which every agent can measure the distance only to his two neighbors, this task is impossible. Subsequently, we propose an algorithm which achieves quiescent but only almost uniform spread. The algorithms we present are scalable and robust. In case the environment (the size of the ring) or the number of agents changes during the run, the swarm adapts and re-deploys without requiring any outside interference. [ABSTRACT FROM AUTHOR]