| Ever since Galois made explicit the connection between the solvability of polynomials and the solvability of groups, the notion has been of interest. A closely related concept is that of a simple group, or the nonexistence of normal subgroups. Both notions have received considerable interest in the 200 years since Galois. In the latter part of the 19th century, the English mathematician William Burnside proved a remarkable theorem: if the order of a group is divisible by only 2 primes, then the group is not simple, and hence solvable. The theorem still stands even after more than a century of other theorems, some requiring 200-page proofs. The proof requires a variety of tools: algebras and their idempotents, association schemes and group characters, and finally some complex numbers. The thesis will discuss each of the areas, and exposes their connections. |
