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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 G be a finite group and let ν(G) be the probability that two randomly selected elements of G produce a nilpotent group. In this article we show that for every positive integer n>0, there is a finite group G such that ν(G)=1n. We also classify all groups G with ν(G)=12. Further, we prove that if G is a solvable nonnilpotent group of even order, then ν(G)p+34p, where p is the smallest odd prime divisor of |G|, and that equality exists if and only if GZ(G) is isomorphic to the dihedral group of order 2p where Z(G) is the hypercenter of G. Finally we find an upper bound for ν(G) in terms of |G| where G ranges over all groups of odd square-free order.

Keywords: Soluble group, Nilpotent subgroup, Probability