One goal of music theory is to describe the resources of a pitch system. Traditionally, the study of pitch intervals was done using frequency ratios of the powers of small integers. Modern mathematical music theory offers an independent way of understanding the pitch system by considering intervals as transformations. This thesis takes advantage of the historical emergence of algebraic structures in musicology and, in the spirit of transformational theory, treats operations that form mathematical groups. Aspects of Neo-Riemannian theory are explored and developed, in particular the T/I and PLR groups as dual. Pitch class spaces, such as 12, can also be defined as torsors. In addition to surveying the group theoretical tools for music analysis, this thesis provides detailed proofs of many claims that are proposed but seldom supported.