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).
This is a preview of subscription content, access via your institution.
References
- 1.
Das, A.K.: On groups elements having square roots. Bull. Iranian Math. Soc. 31(2), 33–36 (2005)
- 2.
Doostie, H., Maghasedi, M.: Certain classes of groups with commutativity degree \(d(G)\le \frac{1}{2}\). Ars. Comb. 89, 263–270 (2009)
- 3.
Erdos, P., Turan, P.: On some problems of a statistical group theory IV. Acta Math. Acad. Sci. Hungar. 19, 413–435 (1968)
- 4.
Guralnick, R.M.: Commutators and commutator subgroups. Adv. Math. 45(3), 319–330 (1982)
- 5.
Gusafson, W.H.: What is the probability that two group elements commute? Am. Math. Monthly. 80, 1031–1034 (1973)
- 6.
Hashemi, M.: The commutativity degree of 2-generated groups of nilpotency class 2. Ars. Comb. 122, 149–159 (2015)
- 7.
Hashemi, M., Pirzadeh, M.: The probability that the commutator equation \([x, y]=g\) has solution in a finite group. J. Algebra Related Topics. 7(2), 47–61 (2019)
- 8.
Hashemi, M., Polkouei, M.: Some numerical results on finite non-abelian \(2\)-generator \(p\)-groups of nilpotency class two. Ann. Math. IASI. LXIII 3, 589–599 (2017)
- 9.
Lucido, M.S., Pournaki, M.R.: Elements with square roots in finite groups. Algebra Colloq. 12(4), 677–690 (2005)
- 10.
Mohd Ali, N.M., Sarmin, N.H.: On some problems in group theory of probabilistic nature. Menemui Matematik. 32(2), 35–41 (2010)
- 11.
Nath, R.K., Prajapati, S.K.: On the number of solutions of a generalized commutator equation in finite groups. Acta Mathematica Hungarica. 156(1), 18–37 (2018)
- 12.
Pournakia, M.R., Sobha, R.: Probability that the commutator of two group elements is equal to a given elemente is equal to a given element. J. Pure Appl. Algebra. 212(4), 727–734 (2008)
- 13.
Sadeghieh, A., Doostie, H.: The \(n^{th}-\)roots of elements in finite groups. Math. Sci. 2(4), 347–356 (2008)
- 14.
Yahya, Z.: The \(n^{th}-\) commutativity degree of some dihedral groups. Menemui Matematik. 34(2), 7–14 (2012)
Author information
Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Keywords
- Groups
- Wreath product
- k ^{th}-roots
- k ^{th}-commutativity degree
Mathematics Subject Classification
- 20P05