An Upper Bound for the Probability of Generating a Finite Nilpotent Group
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)={\f...
Saved in:
Published in | Kyungpook mathematical journal Vol. 63; no. 2; pp. 167 - 173 |
---|---|
Main Authors | , , |
Format | Journal Article |
Language | Korean |
Published |
2023
|
Subjects | |
Online Access | Get full text |
Cover
Loading…
Summary: | 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. |
---|---|
Bibliography: | KISTI1.1003/JNL.JAKO202322252264667 |
ISSN: | 1225-6951 0454-8124 |