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(np). In this paper, we will consider three probabilistic concepts of finite groups. The first problem which we examine is the calculation of the kth-roots of elements in G(np) when \(k\ge 2\). The second problem which is investigated is the computation of the kth-commutative degree of G(np) 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(np).

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Correspondence to Mansour Hashemi.

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Hashemi, M., Pirzadeh, M. & Gorjian, S.A. Some numerical results on the wreath product of \(Z_p\) and \(Z_{p^n}\). Math Sci (2021). https://doi.org/10.1007/s40096-020-00361-6

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Keywords

  • Groups
  • Wreath product
  • k th-roots
  • k th-commutativity degree

Mathematics Subject Classification

  • 20P05