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).

中文翻译：

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关于$$ Z_p $$ Z p和$$ Z_ {p ^ n} $$ Z pn的花圈乘积的一些数值结果

对于每一个正整数Ñ和一个素数p，我们表示的圈积\（Z_p \）和\（Z_ {P ^ N} \）由G ^（Ñ，  p）。在本文中，我们将考虑有限群的三个概率概念。我们研究的第一个问题是当（（k \ ge 2 \）时，计算G（n，  p）中元素的第k^个根。该研究的第二个问题是的计算ķ^个的-commutative度ģ（Ñ，  p）当\（k \ ge 1 \）时。最后，对于\（k \ ge 1 \），我们计算出换向器方程\（[x ^ k，y] = g \）在G（n，  p）中具有解的概率。