Burnside’s Lemma, also referred to as the Cauchy-Frobenius Theorem, is a lemma found in group theory that takes advantage of symmetry in groups to enumerate mathematical objects. Although rooted in group theory, Burnside’s Lemma is also a powerful enumerative combinatorial lemma that allows us to solve counting problems that would otherwise involve tedious casework. In this talk, we will briefly outline the statement and a standard proof of Burnside’s Lemma, go over common problems where the lemma is commonly used, and provide unexpected solutions to problems found in mathematical competitions where the lemma was not intended to be used from the problem author’s perspective.

The competition problems presented are from the AMC and AIME competitions.