The classification of \(p\)-local finite groups over the extraspecial group of order \(p^3\) and exponent \(p\).
Let \(p\) be an odd prime. Influenced by the celebrated work of J. Alperin, \textit{L. Puig} [Structure locale dans les groupes finis, Bull. Soc. Math. Fr., Suppl., Mém. 47 (1976; Zbl 0355.20024)] set out a systematic framework for the study of a finite group in terms of what is known as its \(p\)-local structure or its \(p\)-fusion. This axiomatic development gives a useful tool for determining many theoretic and homotopy theoretic properties of a finite group. The concept of \(p\)-local finite group arose in the paper of \textit{C. Broto, R. Levi} and \textit{R. Oliver} [J. Am. Math. Soc. 16, No. 4, 779-856 (2003; Zbl 1033.55010)] as a generalization of the concept of a finite group and based on Puig's axioms for fusion systems over a \(p\)-group. In this paper, the authors classify the \(p\)-local finite groups over \(p_+^{1+2}\), the extraspecial group of order \(p^3\) and exponent \(p\). This study reduces to the classification of the saturated fusion systems over \(p_+^{1+2}\). As part of this classification, they obtain three new exotic 7-local finite groups. The starting point for this classification is the following fact: if \(G\) is a sporadic finite simple group with Sylow \(p\)-subgroup \(S\) of order \(p^3\), then \(S\) is isomorphic to \(p_+^{1+2}\) and \(p\leq 13\).
scientific article; zbMATH DE number 2104985