Fusion systems on 2-groups with exactly three involutions. (Q2342289)

The author classifies the possible saturated fusion systems on several classes of finite \(2\)-groups with exactly \(3\) involutions. Such groups have been classified by Janko [see \textit{Y. Berkovich} and \textit{Z. Janko}, Groups of prime power order. Vol. 2. de Gruyter Expositions in Mathematics 47. Berlin: Walter de Gruyter (2008; Zbl 1168.20002)] and further studied by \textit{D. A. Craven} and \textit{A. Glesser} [Trans. Am. Math. Soc. 364, No. 11, 5945-5967 (2012; Zbl 1286.20015)]. Using these previous results, the author determines all the saturated fusion systems \(\mathcal F\) on such \(2\)-groups depending on the existence of \(\mathcal F\)-essential subgroups of specific isomorphism type. By Alperin's fusion theorem [e.g. Section 48 of \textit{J. Thévenaz}, \(G\)-algebras and modular representation theory. Oxford: Clarendon Press (1995; Zbl 0837.20015)], a saturated fusion system on a finite \(p\)-group is determined by the \(\mathcal F\)-essential subgroups. The proofs are mainly carried out as a case by case inspection, and complemented using the algebra software GAP. In the last section of the article, the author completes the classification of saturated fusion systems on all finite \(2\)-groups with exactly \(3\) involutions and possessing a non-trivial odd-order automorphism, work which was initiated by Craven and Glesser [loc. cit.].