Counting triangles by group theory

Extract

The introduction of group theory in the sixth form often raises the question of its relationship with other areas of mathematics. The purpose of this note is to develop a group theoretical solution of a simply described counting problem; a problem which may, in fact, be solved in several other ways. The development of the solution yields useful examples of several algebraic notions; in particular the notion of equivalence relation is exploited in a variety of ways. The particular group theoretical/combinatorial result that finally yields the answer to our problem is known as Burnside’s Lemma. This result is not normally presented until a second or third year university course. I hope to convince the reader that the manner of presentation given here is perfectly accessible at a more elementary level.