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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
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References:
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