(University of Birmingham, 2024-06) Alharbi, Faarie; Shpectorov, Sergey

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Axial algebras are commutative, not necessarily associative algebras generated by axes, where axes are idempotents whose adjoint actions are semisimple and such that the product of eigenvectors is controlled by a specific fusion law. Matsuo algebras, defined using Fischer spaces and $3$-transposition groups, are examples of axial algebras of Jordan type $\eta$. In this thesis, we show that Matsuo algebras with $\eta \neq \frac{1}{2}$ almost never contain additional axes, and consequently, their automorphism groups almost always coincide with the automorphism group of the underlying groups of $3$-transpositions. Furthermore, we identify all exceptions to this general rule in the irreducible case. Namely, we find three infinite series and an additional sporadic example of Matsuo algebras having extra axes and a larger group of automorphisms. In particular, for specific $\eta$, we show that $M=M_\eta(S_n)$ is isomorphic to a quotient of $M'=M_\eta(S_{n+1})$, and similarly $M=M_\eta(O^{\epsilon}_{2m}(2))$ is isomorphic to a quotient of $M'=M_\eta(Sp_{2m}(2))$. These are examples of what we call aligned pairs of groups of $3$-transpositions. We use the information about the spectrum of the diagram of irreducible $3$-transposition groups, provided in \cite{JS}, to find all aligned pairs of irreducible $3$-transposition groups. We conclude that, apart from the examples above, there is only one additional sporadic aligned pair $(Fi_{23},\Omega_8(3):S_3)$.