Finite Groups (Sporadic Simple Groups) and Related Topics

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We have studied finite simple groups and related graphs, designs, codes, lattices and vertex operator algebras. Main results are as follows :1. The 2- and 3-radical subgroups of Fischer's simple groups FィイD222ィエD2, FィイD223ィエD2, F'ィイD224ィエD2 have been classified.2. Two kinds of non-split extensions of OィイD27ィエD2(3) have been constructed explicitly by using Dickson's trilinear form and some special Moufang loop.3. By using a lattice VOA, a new proof of the Borwein identity has been given.4. New VOAs related to ternary codes have been introduced, and the irreducible representations of L(ィイD74(/)5ィエD7, 0) 【symmetry】 L(ィイD74(/)5ィエD7, 3) have been determined.5. By using an embedding ィイD82ィエD8AィイD312(/)2ィエD3 into the Leech lattice and a ZィイD22ィエD2 × ZィイD22ィエD2-code, some decomposition of the Moonshine VOA has been given.6. Some 5-designs invariant by PSL(2, 23) and related to S(5, 8, 24) have been classified.7. A simple characterization of S(5, 8, 24) (or Golay code) has been given.8. The Niemeier lattices have been constructed form ZィイD24ィエD2-codes, and their embeddings of them into 1/ィイD82ィエD8 times the Leech lattice have been given.9. The exceptional graphs embedded into the root system EィイD28ィエD2 have been classified.10. Even unimodular Gaussian (resp. Quaternionic) lattices of dimension 12 (resp. 6) have been classified.