Axial algebras are a recently defined type of nonassociative algebras. They are primarily used to realize groups inside the automorphism group of such an algebra. The prominent example is the Griess algebra that was used in 1982 by R.L. Griess to construct the largest sporadic simple group known as the Monster group. Recently, various other groups were realized in this fashion.

In this project, we have introduced modules over axial algebras. If an axial algebra gives rise to a certain group, then the modules of this algebra naturally correspond to representations of (a central extension of) this group.

We will explain the definitions, give examples and provide some insight into recent developments. We will illustrate how the connection between axial algebras and groups can help us to get a better understanding of both. Joint work with Tom De Medts.