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ON THE PROBABILITY OF GENERATING NILPOTENT SUBGROUPS IN A FINITE GROUP
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 93 / Issue 3 / June 2016
- Published online by Cambridge University Press:
- 20 November 2015, pp. 447-453
- Print publication:
- June 2016
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- Article
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- You have access
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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 
${\it\nu}(G)=\frac{1}{12}$.
