### Article

Kyungpook Mathematical Journal 2023; 63(2): 167-173

**Published online** June 30, 2023

Copyright © Kyungpook Mathematical Journal.

### An Upper Bound for the Probability of Generating a Finite Nilpotent Group

Halimeh Madadi, Seyyed Majid Jafarian Amiri, Hojjat Rostami^{*}

Department of Mathematics, Miyaneh Branch, Islamic Azad University, Miyaneh, Iran

e-mail : halime_madadi@yahoo.com

Department of Mathematics, Faculty of Sciences, University of Zanjan, P. O. Box 45371-38791, Zanjan, Iran

e-mail : sm_jafarian@znu.ac.ir

epartment of Mathematics, Farhangian University, Tehran, Iran

e-mail : h.rostami5991@gmail.com

**Received**: February 17, 2022; **Revised**: December 24, 2022; **Accepted**: February 22, 2023

### Abstract

Let

**Keywords**: Soluble group, Nilpotent subgroup, Probability

### 1. Introduction

In the past 40 years, there has been a growing attention in the application of probability in finite groups (for example see [8, 16]). In this paper, we denote by

The notion

Note that for

It is easy to see that

Similarly if

then

However,

A finite group

Gustafson [8] proved that if

In [7] Guralnick and Wilson found that if

It is easy to see that _{5}

Fulman et al. [5] proved that if

**Theorem 1.1.** Suppose that

For a prime _{p}^{k}^{k}

**Conjecture** Let

for some

In Section 2 we compute

In this article

### 2. Computing ν ( G ) for Certain Groups

The following lemmas are very useful in the sequel.

**Lemma 2.1.** Suppose that

**Lemma 2.2.** Suppose that

**Proposition 2.3. ** If

_{1}=h_{2}=1_{1}=k_{2}=1

**Corollary 2.4.** Suppose that ^{r}n

_{2n}

**Corollary 2.5. ** For any integer

In the following we classify all groups

**Proposition 2.6. ** Suppose that

### 3. Upper Bound for ν ( G )

S. Franciosi and F. Giovanni defined and studied a

**Proof of Theorem 1.1.**

If _{i}_{i}_{k}⋉ N

Assume that _{k}

Now we want to count the ordered pairs

Now we continue by induction on the number _{k}

**Case 1**: Assume that

**Case 2**: Suppose that

Let ^{x}=a

Now if _{s}_{s}

For an odd prime

**Corollary 3.1.** Suppose that

**Theorem 3.2. ** Suppose that

for some

_{1}_{p}_{p}

### References

- A. Abdollahi and M. Zarrin,
Non-nilpoten graph of a group , Comm. Algebra,38(12) (2010), 4390-4403. - F. Barry, D. MacHale and A. N She,
Some supersolvability conditions for finite groups , Math. Proc. Royal Irish Acad.,106A(2) (2006), 163-177. - S. Dixmier,
Exposants des quotients des suites centrales descendantes et ascendantes d'un groupe , C. R. Acad. Sci. Paris,259 (1964), 2751-2753. - S. Franciosi and F. Giovanni,
Soluble groups with many nilpotent quotients , Proc. Royal Irish Acad. Sect. A.,89 (1989), 43-52. - J. E. Fulman, M. D. Galloy, G. J. Sherman and J. M. Vanderkam,
Counting nilpotent pairs in nite groups , Ars Combin.,54 (2000), 161-178. - R. M. Guralnick and G. R. Robinson,
On the commuting probability in nite groups , J. Algebra,300(2) (2006), 509-528. - R. M. Guralnick and J. S. Wilson,
The probability of generating a nite soluble group , Proc. London Math. Soc.,81(3) (2000), 405-427. - W. H. Gustafson,
What is the probability that two group elements commute? , Amer. Math. Monthly,80(9) (1973), 1031-1034. - R. Heffernan, D. MacHale and A. N She,
Restrictions on commutativity ratios in finite groups , Int. J. Group Theory,3(4) (2014), 1-12. - S. M. Jafarian Amiri, H. Madadi and H. Rostami,
On the probability of generating nilpotent subgroups in a nite group , Bull. Aust. Math. Soc.,93(3) (2016), 447-453. - E. Khamseh, M. R. R. Moghaddam and F. G. Russo,
Some Restrictions on the Prob- ability of Generating Nilpotent Subgroups , Southeast Asian Bull. Math.,37(4) (2013), 537-545. - P. Lescot, H. N. Nguyen and Y. Yang,
On the commuting probability and supersolv-ability of finite groups , Monatsh. Math.,174 (2014), 567-576. - D. H. McLain,
Remarks on the upper central series of a group , Proc. Glasgow Math. Assoc.,3 (1956), 38-44. - D. J. S. Robinson. A course in the theory of groups. New York: Springer-Verlag; 1996.
- H. Rostami,
The Commutativity Degree in the Class of Nonabelian Groups of Same Order , Kyungpook Math. J.,59 (2019), 203-207. - J. Wilson,
The probability of generating a nilpotent subgroup of a nite group , Bull. London Math. Soc.,40 (2008), 568-580.