Title:

Majorana algebras and subgroups of the Monster

Majorana theory was introduced by A.A. Ivanov as an axiomatisation of certain properties of the 2A axes of the Griess algebra. This work was inspired by that of S. Sakuma who reproved certain important properties of the Monster simple group and the Griess algebra using the framework of vertex operator algebras. The objects at the centre of Majorana theory are known as Majorana algebras and are real, commutative, nonassociative algebras that are generated by idempotents known as Majorana axes. To each Majorana axis, we associate a unique involution in the automorphism group of the algebra, known as a Majorana involution. These involutions form an important link between Majorana theory and group theory. In particular, Majorana algebras can be studied either in their own right or as Majorana representations of finite groups. The main aim of this work is to classify and construct Majorana algebras generated by three axes such that the subalgebra generated by two of these axes is isomorphic to a 2A dihedral subalgebra of the Griess algebra. We first show that such an algebra must occur as a Majorana representation of one of 26 subgroups of the Monster. These groups coincide with the list of trianglepoint subgroups of the Monster given by S. P. Norton. In particular, our result reproves the completeness of Norton's list. This work builds on that of S. Decelle. Next, inspired by work of A. Seress, we design and implement an algorithm to construct the Majorana representations of a given group. We use this to construct a number of important Majorana representations which are independent of the main aim of this work. Finally, we use this algorithm along with our first result to construct all possible Majorana algebras generated by three axes, two of which generate a 2Adihedral algebra. We use these constructions to show that each of these algebras must be isomorphic to a subalgebra of the Griess algebra. This is our main result and can equivalently be thought of as the construction of the subalgebras of the Griess algebra which correspond to the groups in Norton's list of trianglepoint groups.
