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The Conway-Parker algebra and the largest Fischer group
We give a direct, self-contained construction of the three sporadic Fischer groups $\mathrm{Fi}_{24}'$, $\mathrm{Fi}_{23}$, and $\mathrm{Fi}_{22}$ from the $783$-dimensional Conway-Parker algebra. We prove that its distinguished roots define involutory algebra automorphisms whose projective actions
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