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Applying the Kövári-Sós-Turán theorem to a question in group theory. (English) Zbl 07243296
Summary: Let $$m \leq n$$ 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 $$y \in M$$ such that $$\langle x, y \rangle \in \mathfrak X$$. Then either $$G \in \mathfrak X$$ or $$|G| \leq \left(\frac{180}{53}\right)^m (n - 1)$$.
##### MSC:
 20P05 Probabilistic methods in group theory 20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, $$\pi$$-length, ranks 20D99 Abstract finite groups
##### Keywords:
classes of groups; nilpotent groups; soluble groups
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##### References:
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