On the probability of generating a monolithic group

A group L is primitive monolithic if L has a unique minimal normal subgroup, N, and trivial Frattini subgroup. By PL,N(k) we denote the conditional probability that k randomly chosen elements of L generate L, given that they project onto generators for L/N. In this article we show that PL,N(k) is controlled by PY,S(2), where N≅Sr and Y is a 2-generated almost simple group with socle S that is contained in the normalizer in L of the first direct factor of N. Information about PL,N(k) for L primitive monolithic yields various types of information about the generation of arbitrary finite and profinite groups.