The group is a well-known algebraic object and one of the most fundamental structures within mathematics. For a given group, we may correspond the elements to those of a group of invertible matrices, forming a matrix representation of the group. Associated with these representations are characters, obtained by taking the traces of each matrix. Studying these characters, we find that they contain much information about their underlying group. One of the main goals in character theory is to determine what properties about the group can be determined by looking at its characters. We will discuss several recent conjectures on this topic that study properties of certain subgroups known as defect groups. In particular, we prove some such conjectures regarding generation properties of defect 2-groups for the symplectic group Sp4(q), which is a particular group of 4 & times; 4 matrices over any finite field Fq of odd characteristic.
