Large product-free subsets of finite groups
A subset of a group is said to be product-free if the product of two of its elements is never itself an element of the subset. Let \(G\) be a finite group of order \(n>1\). Define \(\alpha(G)\) to be the size of the largest product-free subset of \(G\). \textit{L. Babai} and \textit{V. T. Sós} [Eur. J. Comb. 6, 101-114 (1985; Zbl 0573.05032)] gave a simple construction that, together with the classification of finite simple groups, shows that \(\alpha(G)>cn^{4/7}\) for some constant \(c>0\). In the paper under review the author proves that \(\alpha(G)>cn^{11/14}\) for some constant \(c>0\). Since \(\alpha(G)\geq|N|\alpha(G/N)\) for any proper normal subgroup \(N\) of \(G\) (Lemma 1), it suffices to consider the finite simple groups. Let \(P(G)\) denote the index of the largest proper subgroup of \(G\). Then there exists \(c>0\) such that \(P(G)<c|G|^{3/7}\) for all nonabelian simple groups \(G\) (Lemma 2). As to \(P(G)\), the known results for the classical and exceptional families are summarized in Tables I and II. (Table II is culled from a recent paper of \textit{M. W. Liebeck} and \textit{A. Shalev} [J. Algebra 184, No. 1, 31-57 (1996; Zbl 0870.20014)].) Babai and Sós observed that \(\alpha(G)\geq|G|/P(G)\). By a probabilistic method, the author obtains that \(\alpha(G)\geq|G|(31P(G))^{-1/2}\) (Lemma 3).
scientific article; zbMATH DE number 980669