Embeddings of Simple Groups in Exceptional Finite Simple Groups

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This project will investigate the subgroups of the finite exceptional simple group E_8(q) for small values of q (q a power of a prime). One of the main focuses will be the calculation of Brauer characters for both E_8(q) and also for various other simple groups H. It will be important here to have knowledge of conjugacy classes, particularly of E_8(q). The aim is to see which admissible fusions and restrictions of H on the 248 dimensional GF(q) module forE_8(q) can occur. This will give information on possible embeddings of H in E_8(q) and in some cases will show that such an H cannot be embedded. Calculating the values of Brauer characters of large order elements can some times be problematic. These groups are very large - for example E_8(3) has approximately 10^{123} elements and it is expected that computer algebra will be heavily involved in this project, particularly indetermining the possible admissible embeddings.EPSRC research area: Mathematics/Algebra