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

Khaled Alajmi

doi:doi:10.11648/j.acm.20251404.13

Research Article |

| Peer-Reviewed

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

Khaled Alajmi^*

Published in Applied and Computational Mathematics (Volume 14, Issue 4)

Received: 21 June 2025 Accepted: 7 July 2025 Published: 5 August 2025

Views: Downloads:

Download PDF

Share This Article

Twitter
Linked In
Facebook

Abstract

Let G be a finite group. Define λ_n(G) to be the probability that n elements drawn at random with replacement from G generate G. Define E(G) to be the expected number of elements of G which have to be drawn at random with replacement from G before a set of generators is found. The purpose of this paper is to compute λ_n(G) and E(G) for certain finite nilpotent groups including non-abelian groups. In this paper we have, in particular, computed λ_n(G) as a first step then E(G) for the groups G where G is a nilpotent group isomorphic to the direct product of its pi-Sylow subgroups, for cyclic groups ℤ_q, q is a power of a prime p and for non-abelian groups of order p⁴ of the shape ℤ_p² ⋊ ℤ_p² the semi-direct product of two copies of ℤ_p². 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.

Published in	Applied and Computational Mathematics (Volume 14, Issue 4)
DOI	10.11648/j.acm.20251404.13
Page(s)	210-215
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2025. Published by Science Publishing Group

Keywords

Probability, Expectation, Nilpotent Groups, p−groups, Euler Function

References

[1]	K. Alajmi, On expectation of generating certain finite nilpotent groups, Journal of Applied Mathematics and Computation, 2024, 8(2), 113-119.
[2]	V. Acciaro, The probability of generating some common families of finite groups. Utilitas Mathematica, 49, (1996), 243-253.
[3]	B. Alhasanat, On classification of groups of order p⁴, p odd prime, International Journal of Mathematics and Computer Science, 17 No. 4, (2022), 1564-1593.
[4]	B. Alhasanat, Some probabilistic approaches on finite groups, International Journal of Applied Mathematics, 35 No. 6, (2022), 827-838.
[5]	Y. Anotolin, A. Martino, E. Ventura, degree of commutativity of finite groups, Proc. Amer. Math. Soc. 145 (2017), 479-485. https://doi.org/10.48550/arXiv.1511.07269
[6]	M. Atkinson, A survey of algorithms for handling permutation groups. School of Computer Science. Technical report SCS-TR-164, Carlton University, Ottawa, January. (1990).
[7]	T. Burness, R. Guralnik, A. Morito, G. Navarro, On the commuting probability of p-elements in a finite group, Algebra and number theory, 17(6) (2023), 1209-1229. https://doi.org/10.2140/ant.2023.17.1209
[8]	A. Chashiani, R. Rezaie, On the commutativity degree of a group algebra, Africa Mathematica, 37 (2021) 1137-1145.
[9]	J. D. Dixon, Problems in group theory. Blaisdell Publishing Company, 1967.
[10]	S. Eberhard, Pavel Shumyatsky. Probabilistically nilpotent groups of class 2. Mathematische Annalen, https://doi.org/10.1007/500208 .023.02567.0 Accepted and to appear, (2023).
[11]	A. Erfanian, R. Rezaie, P.Lescot, On the relative commutativity degree of a subgroup of a finite group. Communications in Algebra 35 (2007), 4183-4197 https://doi.org/10.1080/00927870701545044
[12]	M. Farrokhi D. G., Finite groups with five relative Commutativity degrees, Results in Mathematics, 77(2022), article number 56. https://doi.org/10.48550/arXiv.1912-04550
[13]	M. Hashimi, An extension of commutativity degree of finite groups, Journal of algebra and related topics, https://doi.org/10.22124/JART2024.26440.1616
[14]	P. Hall, The Eulerian functions of a group. Quart. J.Math., Ox. Series 7, (1936), 134-151.
[15]	R. Kantinath, Commutativity degree of class of finite groups and consequences, Bull. Aust.Math. Soc., 85 (2013), 448-452.
[16]	P. Lescot, Central extension and commutativity degree, Communications in Algebra, 29 (10) (2001), 4451-4460. https://doi.org/10.1081/AGB-100106768
[17]	H. Madadi, S. Amiri, H. Rostami, An upper bound for the probability of generating a finite nilpotent group, Kyungpook Math. J. 63, (2023), 167-173. https://doi.org/10.5666/KMJ.2023.63.2.167

Cite This Article

Plain Text BibTeX RIS

APA Style

Alajmi, K. (2025). The Expectation of Generating Certain Finite Nilpotent Groups and Non-abelian Groups. Applied and Computational Mathematics, 14(4), 210-215. https://doi.org/10.11648/j.acm.20251404.13

Copy | Download

ACS Style

Alajmi, K. The Expectation of Generating Certain Finite Nilpotent Groups and Non-abelian Groups. Appl. Comput. Math. 2025, 14(4), 210-215. doi: 10.11648/j.acm.20251404.13

Copy | Download

AMA Style

Alajmi K. The Expectation of Generating Certain Finite Nilpotent Groups and Non-abelian Groups. Appl Comput Math. 2025;14(4):210-215. doi: 10.11648/j.acm.20251404.13

Copy | Download

@article{10.11648/j.acm.20251404.13,
  author = {Khaled Alajmi},
  title = {The Expectation of Generating Certain Finite Nilpotent Groups and Non-abelian Groups
},
  journal = {Applied and Computational Mathematics},
  volume = {14},
  number = {4},
  pages = {210-215},
  doi = {10.11648/j.acm.20251404.13},
  url = {https://doi.org/10.11648/j.acm.20251404.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20251404.13},
  abstract = {Let G be a finite group. Define λn(G) to be the probability that n elements drawn at random with replacement from G generate G. Define E(G) to be the expected number of elements of G which have to be drawn at random with replacement from G before a set of generators is found. The purpose of this paper is to compute λn(G) and E(G) for certain finite nilpotent groups including non-abelian groups. In this paper we have, in particular, computed λn(G) as a first step then E(G) for the groups G where G is a nilpotent group isomorphic to the direct product of its pi-Sylow subgroups, for cyclic groups ℤq, q is a power of a prime p and for non-abelian groups of order p4 of the shape ℤp2 ⋊ ℤp2 the semi-direct product of two copies of ℤp2. 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.
},
 year = {2025}
}

Copy | Download

TY  - JOUR
T1  - The Expectation of Generating Certain Finite Nilpotent Groups and Non-abelian Groups

AU  - Khaled Alajmi
Y1  - 2025/08/05
PY  - 2025
N1  - https://doi.org/10.11648/j.acm.20251404.13
DO  - 10.11648/j.acm.20251404.13
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 210
EP  - 215
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20251404.13
AB  - Let G be a finite group. Define λn(G) to be the probability that n elements drawn at random with replacement from G generate G. Define E(G) to be the expected number of elements of G which have to be drawn at random with replacement from G before a set of generators is found. The purpose of this paper is to compute λn(G) and E(G) for certain finite nilpotent groups including non-abelian groups. In this paper we have, in particular, computed λn(G) as a first step then E(G) for the groups G where G is a nilpotent group isomorphic to the direct product of its pi-Sylow subgroups, for cyclic groups ℤq, q is a power of a prime p and for non-abelian groups of order p4 of the shape ℤp2 ⋊ ℤp2 the semi-direct product of two copies of ℤp2. 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.

VL  - 14
IS  - 4
ER  -

Copy | Download

Author Information

Khaled Alajmi

Department of Mathematics, College of Basic Education, Public Authority for Applied Education and Training, Ardiyah, Kuwait

Contact Email

Download PDF

Submit an Article

Sections

Plain Text BibTeX RIS

APA Style

Alajmi, K. (2025). The Expectation of Generating Certain Finite Nilpotent Groups and Non-abelian Groups. Applied and Computational Mathematics, 14(4), 210-215. https://doi.org/10.11648/j.acm.20251404.13

Copy | Download

ACS Style

Alajmi, K. The Expectation of Generating Certain Finite Nilpotent Groups and Non-abelian Groups. Appl. Comput. Math. 2025, 14(4), 210-215. doi: 10.11648/j.acm.20251404.13

Copy | Download

AMA Style

Alajmi K. The Expectation of Generating Certain Finite Nilpotent Groups and Non-abelian Groups. Appl Comput Math. 2025;14(4):210-215. doi: 10.11648/j.acm.20251404.13

Copy | Download

@article{10.11648/j.acm.20251404.13,
  author = {Khaled Alajmi},
  title = {The Expectation of Generating Certain Finite Nilpotent Groups and Non-abelian Groups
},
  journal = {Applied and Computational Mathematics},
  volume = {14},
  number = {4},
  pages = {210-215},
  doi = {10.11648/j.acm.20251404.13},
  url = {https://doi.org/10.11648/j.acm.20251404.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20251404.13},
  abstract = {Let G be a finite group. Define λn(G) to be the probability that n elements drawn at random with replacement from G generate G. Define E(G) to be the expected number of elements of G which have to be drawn at random with replacement from G before a set of generators is found. The purpose of this paper is to compute λn(G) and E(G) for certain finite nilpotent groups including non-abelian groups. In this paper we have, in particular, computed λn(G) as a first step then E(G) for the groups G where G is a nilpotent group isomorphic to the direct product of its pi-Sylow subgroups, for cyclic groups ℤq, q is a power of a prime p and for non-abelian groups of order p4 of the shape ℤp2 ⋊ ℤp2 the semi-direct product of two copies of ℤp2. 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.
},
 year = {2025}
}

Copy | Download

TY  - JOUR
T1  - The Expectation of Generating Certain Finite Nilpotent Groups and Non-abelian Groups

AU  - Khaled Alajmi
Y1  - 2025/08/05
PY  - 2025
N1  - https://doi.org/10.11648/j.acm.20251404.13
DO  - 10.11648/j.acm.20251404.13
T2  - Applied and Computational Mathematics
JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
SP  - 210
EP  - 215
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20251404.13
AB  - Let G be a finite group. Define λn(G) to be the probability that n elements drawn at random with replacement from G generate G. Define E(G) to be the expected number of elements of G which have to be drawn at random with replacement from G before a set of generators is found. The purpose of this paper is to compute λn(G) and E(G) for certain finite nilpotent groups including non-abelian groups. In this paper we have, in particular, computed λn(G) as a first step then E(G) for the groups G where G is a nilpotent group isomorphic to the direct product of its pi-Sylow subgroups, for cyclic groups ℤq, q is a power of a prime p and for non-abelian groups of order p4 of the shape ℤp2 ⋊ ℤp2 the semi-direct product of two copies of ℤp2. 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.

VL  - 14
IS  - 4
ER  -

Copy | Download