Block fusion systems and the center of the group ring

This thesis develops some aspects of the theory of block fusion systems. Chapter 1 contains a brief introduction to the group algebra and some simple results about algebras over a field of positive characteristic. In chapter 2 we define the concept of a fusion system and the fundamental property of saturation. We also define block fusion systems and prove that they are saturated. Chapter 3 develops some tools for relating block fusion systems to the structure of the center of the group algebra. In particular, it is proven that a block has trivial defect group if and only if the center of the block algebra is one-dimensional. Chapter 4 consists of a proof that block fusion systems of symmetric groups are always group fusion systems of symmetric groups, and an analogous result holds for the alternating groups.