ABSTRACT
This chapter contains an introduction to some aspects of group theory that are directly related to combinatorial problems. The first part of the chapter gives the basic definitions of group theory and derives some fundamental properties of symmetric groups. We apply this material to give combinatorial derivations of the basic properties of determinants. The second part of the chapter discusses group actions, which have many applications to algebra and combinatorics. In particular, group actions can be used to solve counting problems in which symmetry must be taken into account. For example, how many ways can we color a 5×5 chessboard with seven colors, if all rotations and reflections of a given colored board are considered the same? The theory of group actions provides systematic methods for solving problems like this one.
