Applying the Kövári-Sós-Turán theorem to a question in group theory

Abstract

Let $m\le n$ be positive integers and $\mathfrak{X}$ a class of groups which is closed for subgroups, quotient groups and extensions. Suppose that a finite group G satisfies the condition that for every two subsets M and N of cardinalities m and n, respectively, there exist $x\in M$ and $y\in N$ such that $〈x,y〉\in \mathfrak{X}.$ Then either $G\in \mathfrak{X}$ or $\left|G\right|\le {\left(\frac{180}{53}\right)}^m(n-1).$

Keywords:

• Classes of groups
• nilpotent groups
• soluble groups

MATHEMATICS SUBJECT CLASSIFICATION:

• 20P05
• 20D10
• 20D99