Abstract. Axial algebras are a class of non-associative commutative algebras whose properties are defined in terms of a fusion law. When this fusion law is 2-graded, the algebra has a naturally associated automorphism group generated by involutions, and thus axial algebras are related to the theory of finite simple groups. Examples of axial algebras include finite-dimensional simple Jordan algebras and the Grice algebra. In this talk, we generalize the theory of axial algebras and construct graphs whose automorphism
groups are finite almost simple groups.
