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In this dissertation, we compute the essential p-dimension of the split finite quasi-simple groups of classical Lie type at the defining prime, specifically the quasi-simple groups arising from the general linear and special linear groups, the symplectic groups, and the orthogonal groups. Also, for odd primes l not equal to the defining prime, we compute the essential l-dimension of the finite groups of classical Lie type, specifically the general linear and special linear groups, the symplectic groups, the orthogonal groups, and the unitary groups, and the non-abelian simple factors in their Jordan-Hölder series.

Abstract: When computing the essential dimension of an algebraic group G defined over a field k, finding lower bounds is generally a much more difficult problem than finding upper bounds. For simple algebraic groups G of adjoint type, Chernousov-Serre developed a general method for computing lower bounds of G via an orthogonal representation. Their work did not cover the case when char(k) = 2, but they did note their belief that the method could be extended to this case. We will show that the method developed by Chernousov-Serre does indeed work in the characteristic 2 case. As an application, we employ the method to assist with the computation of the essential dimension of the orthogonal group O_n and simple adjoint groups of type G_2 in the characteristic 2 case.

Interest in essential dimension problems has been growing over the past decade. In part, it is because the idea of essential dimension captures quite elegantly the problem of parametrizing a wide range of algebraic objects. But perhaps more, it is because the study of essential dimension requires most of the algebraic arsenal. What began as a problem in Galois cohomology and representation theory now has connections to versal torsors, stacks, motives, birational geometry, and invariant theory. This exposition will focus on just a small bit of this theory: algebraic tori and how they can be used to help us calculate the essential p-dimension for PGLn.

In this thesis we study the essential dimension of the first Galois cohomology functors of finite groups. Following the result by N. A. Karpenko and A. S. Merkurjev about the essential dimension of finite p-groups over a field containing a primitive p-th root of unity, we compute the essential dimension of finite cyclic groups and finite abelian groups over a field containing all primitive p-th roots of unity for all prime divisors p of the order of the group. For the computation of the upper bound of the essential dimension we apply the techniques about affine group schemes, and for the lower bound we make use of the tools of canonical dimension and fibered categories. We also compute the essential dimension of small groups over the field of rational numbers to illustrate the behavior of essential dimension when the base field does not contain all relevant primitive roots of unity.