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ON THE PROBABILITY OF GENERATING NILPOTENT SUBGROUPS IN A FINITE GROUP

Part of: Probabilistic methods in group theory Representation theory of groups

Published online by Cambridge University Press:  20 November 2015

S. M. JAFARIAN AMIRI ,
H. MADADI  and
H. ROSTAMI
S. M. JAFARIAN AMIRI*
Affiliation:
Department of Mathematics, Faculty of Sciences, University of Zanjan, PO Box 45371-38791, Zanjan, Iran email [email protected]
H. MADADI
Affiliation:
Department of Mathematics, Faculty of Sciences, University of Zanjan, PO Box 45371-38791, Zanjan, Iran email [email protected]
H. ROSTAMI
Affiliation:
Department of Mathematics, Faculty of Sciences, University of Zanjan, PO Box 45371-38791, Zanjan, Iran email [email protected]
*
✉[email protected]
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Abstract

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Let $G$ be a finite group. We denote by ${\it\nu}(G)$ the probability that two randomly chosen elements of $G$ generate a nilpotent subgroup and by $\text{Nil}_{G}(x)$ the set of elements $y\in G$ such that $\langle x,y\rangle$ is a nilpotent subgroup. A group $G$ is called an ${\mathcal{N}}$-group if $\text{Nil}_{G}(x)$ is a subgroup of $G$ for all $x\in G$. We prove that if $G$ is an ${\mathcal{N}}$-group with ${\it\nu}(G)>\frac{1}{12}$, then $G$ is soluble. Also, we classify semisimple ${\mathcal{N}}$-groups with ${\it\nu}(G)=\frac{1}{12}$.

Keywords

nilpotent groupprobabilityN-group

MSC classification

Primary: 20D60: Arithmetic and combinatorial problems
Secondary: 20P05: Probabilistic methods in group theory
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 93 , Issue 3 , June 2016 , pp. 447 - 453
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

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