Abstract
The sporadic simple group Monster, denoted by M, acts on the Griess algebra, which is a real vector space of dimension 196,884, equipped with a positive definite scalar product and a bilinear, commutative, and non-associative algebra product. Certain properties of this linear representation of M, together with properties (discovered by Conway and Miyamoto) of idempotents in the Griess algebra that correspond to 2A involutions in M, have been defined by Ivanov as the M-representation of the Monster. This definition enables us to talk about M-representations of arbitrary groups G that are generated by involutions. In general, an M-representation may or may not exist, but if G is isomorphic to a subgroup of the Monster and a representation is isomorphic to the corresponding subalgebra of the Griess algebra then we say that the M-representation is based on an embedding of G in the Monster.
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