In this paper , we consider the probability that two elements chosen at random from a finite group G generate a subgroup of a given nilpotency class. It is shown that in solvable non-nilpotent groups, the probability that two elements generate a nilpotent subgroup is <= l/p,, where p, is the smallest prime dividing the order of the group, and it is also shown that there exist groups such that the probability of two elements generating a subgroup of class i approaches one (and other groups for which it approaches zero) for all i =>2. It is also shown that the number of pairs which generate a subgroup of a given class is always a multiple of the order of the group. Some preliminary results on the analogous problem for solvability are also given.
Fulman, Jason; Galloy, Michael; and Vanderkam, Jeffery, "Counting Nilpotent Pairs" (1992). Mathematical Sciences Technical Reports (MSTR). 135.