### Article

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

**Published online** June 30, 2023 https://doi.org/10.5666/KMJ.2023.63.2.167

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}

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