The probability of generating finite and profinite groups

Famously, every finite simple group can be generated by a pair of elements. Moreover, Liebeck and Shalev (1995) proved that the probability that a pair of elements generate tends to 1 as . In this paper, we generalize this theorem of Liebeck and Shalev. Work of Lucchini and Menegazzo (1997) implies that a finite group can be generated by a pair of elements if it has a unique chief series. As a consequence of our main theorem, the probability that a pair of elements generate such a group tends to 1 as , where is the unique simple quotient of . We also prove that a profinite group with finitely many chief series has a topological generating set of size , and for any such , the probability that a ‐tuple of elements topologically generates is positive; moreover, we can take if has a unique chief series. Along the way, we show that the chief factors of a finite group with a unique chief series are highly constrained, and we also analyze the maximal subgroup zeta function of a finite group with a unique minimal normal subgroup.