The Expectation of Generating Certain Finite Nilpotent Groups and Non-abelian Groups

Let G be a finite group. Define λn(G) to be the probability that n elements drawn at random with replacement from G generate G. Define E(G) to be the expected number of elements of G which have to be drawn at random with replacement from G before a set of generators is found. The purpose of this paper is to compute λn(G) and E(G) for certain finite nilpotent groups including non-abelian groups. In this paper we have, in particular, computed λn(G) as a first step then E(G) for the groups G where G is a nilpotent group isomorphic to the direct product of its pi-Sylow subgroups, for cyclic groups ℤq, q is a power of a prime p and for non-abelian groups of order p⁴ of the shape ℤp² ⋊ ℤp² the semi-direct product of two copies of ℤp². These results are knew and could lead to give some alternative description of the structure of the group and its elements. In general probabilistic group theory has applications on probabilistic methods to prove deterministic theorems in group theory.