The Art of Problem Solving has just announced a new course starting this fall in Group Theory. Like all of the AOPS courses this is sure to be an excellent offering. Here is the AOPS course description:
Group theory is the study of symmetry. Objects in nature (math, physics, chemistry, etc.) have beautiful symmetries and group theory is the algebraic language we use to unlock that beauty. This example-driven course will focus specifically on building groups from other groups, ways that groups can act on various sets, and using the tools of group theory to construct fields.
Group theory is the gateway to abstract algebra, one of the two core branches of higher mathematics. (Just as calculus is the gateway to the other core branch, analysis.) Algebra and analysis together hold the tools for most areas of advanced mathematics, from geometry to topology to applied mathematics and beyond. Algebra is what tells us (among many other things) that you can’t trisect an angle, that there are finitely many regular polyhedra, and that there is no closed form for solving a quintic, for example. In this class we will get a glimpse of the mathematics underlying these famous questions.
Note: We will not assume prior knowledge of what a group is. Group theory is a topic that is generally reserved for a sophomore or junior level abstract algebra course, but this class attempts to present the most important concepts and examples on a level appropriate for very strong high school students. The goal of this class, as with all of our classes, is to provide a much richer curriculum that challenges students who are otherwise exhausting the subjects available to them.