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

Let iG/i be a finite group. Define iλsubn/sub(G)/i to be the probability that in/i elements drawn at random with replacement from iG/i generate iG/i. Define iE(G)/i to be the expected number of elements of iG/i which have to be drawn at random with replacement from iG/i before a set of generators is found. The purpose of this paper is to compute iλsubn/sub(G)/i and iE(G)/i for certain finite nilpotent groups including non-abelian groups. In this paper we have, in particular, computed iλsubn/sub(G)/i as a first step then iE(G)/i for the groups iG/i where iG/i is a nilpotent group isomorphic to the direct product of its pi-Sylow subgroups, for cyclic groups ℤsubq/sub, iq/i is a power of a prime ip/i and for non-abelian groups of order ip/isup4/sup of the shape ℤsubp/subsup2/sup ⋊ ℤsubp/subsup2/sup the semi-direct product of two copies of ℤsubp/subsup2/sup. 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.