About Group Theory
A group is the collection of symmetries of an object. In the study of group theory, there is an information exchange between a group and the object on which the group acts. A modern approach comes from considering the object to be the group itself, but as either a geometric or a measure-theoretic object. The result is an interchange between the algebraic information of a group and its geometric and measure-theoretic counterparts. The class of finitely generated groups provides a rich and natural source of examples for this framework, yielding an expansive field of research that touches upon many beautiful areas of mathematics.