Finite groups with a certain number of elements pairwise generating a non-nilpotent subgroup

Abstract

Let $n>0$ be an integer and $\mathcal{X}$ be a class of groups. We say that a group $G$ satisfies the condition $(\mathcal{X},n)$ whenever in every subset with $n+1$ elements of $G$ there exist distinct elements $x,y$ such that $<x,y>$ is in $\mathcal{X}$. Let $\mathcal{N}$ and $ \mathcal{A}$ be the classes of nilpotent groups and abelian groups, respectively. Here we prove that: (1) If $G$ is a finite semi-simple group satisfying the condition $(\mathcal{N},n)$, then $|G|<c^{2[\log_{21}n]n^2} [\log_{21}n]!$, for some constant $c$. (2) A finite insoluble group $G$ satisfies the condition $(\mathcal{N},21)$ if and only if $\frac{G}{Z^*(G)}\cong A_5$, the alternating group of degree 5, where $Z^*(G)$ is the hypercentre of $G$. (3) A finite non-nilpotent group $G$ satisfies the condition $(\mathcal{N}, 4)$ if and only if $\frac{G}{Z^*(G)}\cong S_3$, the symmetric group of degree 3. (4) An insoluble group $G$ satisfies the condition $(\mathcal{A},21)$ if and only if $G\cong Z(G)\times A_5$, where $Z(G)$ is the centre of $G$. (5) If $d$ is the derived length of a soluble group satisfying the condition $(\mathcal{A},n)$, then $d=1$ if $n\in \{1,2\}$ and $d\leq 2n-3$ if $n\geq 2$.