Abstract. In this article, we study Griess algebras and vertex operator subalgebras generated by Ising vectors in a moonshine type VOA such that the subgroup generated by the corresponding Miyamoto involutions has the shape 3 2 :2 and any two Ising vectors generate a 3C subVOA U 3C . We show that such a Griess algebra is uniquely determined, up to isomorphisms. The structure of the corresponding vertex operator algebra is also discussed. In addition, we give a construction of such a VOA inside the lattice VOA, which gives an explicit example for Majorana representations of the group 3 2 :2 of 3C-pure type.

In this article, we first compute the quantum dimensions of irreducible V τ √ 2A 2 -modules. These quantum dimensions give upper bounds on fusion rules among irreducible V τ √ 2A 2 -modules. Together with the lower bounds obtained by Lam and the author, we determine explicitly fusion rules among all irreducible V τ √ 2A 2 -modules of twisted type. This work completes the program for determining the fusion rules among irreducible V τ √ 2A 2 -modules.

Abstract. In this article, we construct explicitly certain moonshine type vertex operator algebras generated by a set of Ising vectors I such that (1) for any e = f ∈ I, the subVOA VOA(e, f ) generated by e and f is isomorphic to either U 2B or U 3C ; and (2) the subgroup generated by the corresponding Miyamoto involutions {τ e | e ∈ I} is isomorphic to the Weyl group of a root system of type A n , D n , E 6 , E 7 or E 8 .The structures of the corresponding vertex operator algebras and their Griess algebras are also studied. In particular, the central charge of these vertex operator algebras are determined.

In this article, we describe some maximal 3-local subgroups of the Monster simple group using vertex operator algebras (VOA). We first study the holomorphic vertex operator algebra obtained by applying the orbifold construction to the Leech lattice vertex operator algebra and a lift of a fixed-point free isometry of order 3 of the Leech lattice. We also consider some of its special subVOAs and study their stabilizer subgroups using the symmetries of the subVOAs. It turns out that these stabilizer subgroups are 3-local subgroups of its full automorphism group. As one of our main results, we show that its full automorphism group is isomorphic to the Monster simple group by using a 3-local characterization and that the holomorphic VOA is isomorphic to the Moonshine VOA. This approach allows us to obtain relatively explicit descriptions of two maximal 3-local subgroups of the shape 3 1+12 .2. Suz :2 and 3 8 .Ω − (8, 3).2 in the Monster simple group.

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