Applications of the classification of finite simple groups and prime graphs

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We studied about the structures of finite simple groups and found several properties of finite groups using the classification of finite simple groups.Using properties of solvable graphs of finite groups Iiyori generalized P. Hall's theorem on the solvability of finite groups.Sawabe constructed a new p-local geometry which includes several geometries on the sporadic simple groups. He proved that the reduced Lefschetz module of the G-poset consisting of all centric p-radical subgroups of a finite group G is an χ-projective virtual Zp[G]-module where χ is a family of p-subgroups of the normalizers of the non-centric p-radical subgroups of G.Chigira studied Bender-Glauberman's idea on the proof of Burnside's conjecture and simplified Suzuki's last paper using Peterfalvi's character theory for the groups of odd order. Chigira also gave an elementary construction of Mathieu simple groups M_<11> and M_<12>.Endo gave a generalization of the Maillet determinant and the Demyanenko matrix.