Automorphism groups of axial algebras

Axial algebras are a class of commutative non-associative algebras which have a natural group of automorphisms, called the Miyamoto group. The motivating example is the Griess algebra which has the Monster sporadic simple group as its Miyamoto group. Previously, using an expansion algorithm, about 200 examples of axial algebras in the same class as the Griess algebra have been constructed in dimensions up to about 300. In this list, we see many reoccurring dimensions which suggests that there may be some unexpected isomorphisms. Such isomorphisms can be found when the full automorphism groups of the algebras are known. Hence, in this paper, we develop methods for computing the full automorphism groups of axial algebras and apply them to a number of examples of dimensions up to 151.