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

• Published in: Kyungpook mathematical journal Vol. 63; no. 2; pp. 167 - 173
• Main Authors: Halimeh Madadi, Seyyed Majid Jafarian Amiri, Hojjat Rostami
• Format: Journal Article
• Language: Korean
• Published: 2023
• Subjects: Nilpotent subgroup; Soluble group; Probability
• ISSN: 1225-6951

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 ${\nu}(G)={\frac{1}{n}}$. We also classify all groups G with ${\nu}(G)={\frac{1}{2}}$. Further, we prove that if G is a solvable nonnilpotent group of even order, then ${\nu}(G){\leq}{\frac{p+3}{4p}}$, where p is the smallest odd prime divisor of |G|, and that equality exists if and only if $\frac{G}{Z_{\infty}(G)}$ 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.