Every finite simple group is generated by two elements, and this Idea studies how strongly that holds. Random generation: P_k(G) is the probability that k uniformly random elements generate G, and Dixon's theorem gives P_2(G) -> 1 along the finite simple groups, quantified by the subgroup zeta function zeta_G(s) = sum over maximal subgroups H of |G:H|^{-s}. (a,b)-generation asks for a generating pair consisting of an element of order a and an element of order b; (2,3)-generation identifies the group as a quotient of the modular group PSL(2,Z), and triangle generation by a hyperbolic triple (a,b,c) realises it as a quotient of a Fuchsian group, the Hurwitz case (2,3,7) meeting Hurwitz's bound of 84(g-1) automorphisms of a compact Riemann surface of genus g. Spread: G has spread k if for any k non-identity elements x_1..x_k there is a single y with <x_i,y> = G for every i, and uniform spread k strengthens this by drawing y from one fixed conjugacy class; every finite simple group has uniform spread at least 2, proved by bounding fixed point ratios fpr(x,G/H) = |x^G intersect H| / |x^G| over maximal subgroups. The generating graph, whose vertices are the non-identity elements joined when they generate G, encodes spread as connectivity and is conjecturally Hamiltonian for large simple groups. Generating subgroups: bounds on the minimal number of generators d(H) for subgroups H of a simple group, on the number of subgroups needed to cover or generate, and on maximal and second maximal subgroups, using the O'Nan-Scott theorem, Aschbacher's subgroup structure theorem, Zsigmondy primitive prime divisors, and the classification of finite simple groups.
A survey by Timothy C. Burness, based on his one-hour lecture at Groups St Andrews 2017, on the generation properties of finite simple groups and on the probabilistic techniques used to prove them. I…