Group Actions and Invariants

Why are the trace and the determinant of a matrix special, as opposed to some other random function of the matrix entries? Why should the discriminant $b^2-4ac$ play a fundamental role, and not, say $a^2+4bc$? One answer (among many) is that the trace, the determinant, and the discriminant are distinguished among other functions by the fact that they are invariant functions under a natural group action. We will explore group actions and their invariants in algebra, representation theory, and geometry. Projects can be expository or include original research.