This is a project in geometric group theory, which is the study of groups by investigating connections between algebraic properties of groups and geometric properties of spaces. A group is a set of elements with an operation which combines any two elements to form another element. For example, the special linear group of degree 2 is a set of 2×2 matrices which can perform translations in a coordinate plane, but still preserves the orientation and area of the original figure. In this project, we investigated various examples of finite groups and infinite groups, such as the set of integers, special linear groups, and Braid groups. We then explored their geometrical representations, namely in Cayley graphs. Students learned how to prove mathematical statements and expose themselves to different levels of algebra.