Some numerical results on the wreath product of $Z_p$ and $Z_{p^n}$

Abstract

For every positive integer n and for a prime number p, we denote the wreath product of $Z_p$ and $Z_{p^n}$ by G(n, p). In this paper, we will consider three probabilistic concepts of finite groups. The first problem which we examine is the calculation of the k^th-roots of elements in G(n, p) when $k\ge 2$. The second problem which is investigated is the computation of the k^th-commutative degree of G(n, p) when $k\ge 1$. In the end, for $k\ge 1$ we compute the probability that the commutator equation $[x^k,y]=g$ has solution in G(n, p).