A Simple Proof of Rankin's Campanological Theorem
Richard G. Swan · American Mathematical Monthly · 1999
Change ringing is the traditional English method of ringing church bells. The basic idea is to ring a set of bells in all possible orders (the changes) with no repetition until the initial position recurs. The sequence of changes is usually grouped into blocks, known as 'leads,' of a standard form, and one considers the sequence consisting of the last change in each lead (the 'lead ends'). Each lead end is obtained from the previous one by a permutation depending on the type of lead, and one tries to choose the sequence of leads so that all possible changes occur. The mathematical problem involved in doing this can be formulated more generally as follows. Given a finite group G with a set of generators, E, one attempts to enumerate the elements of G as x1, ... , x,, (with n = I G I) in such a way that for each i, xi,1 = xiei for some ei in E (including xl = xne). Many explicit solutions have been given in particular cases, often by quite ingenious methods [5], [6], but few general results seem to be known about the possibility of constructing such a sequence. Aside from the obvious requirement that E generates G, the only necessary condition known to me is a theorem of Rankin [4], which generalizes an earlier result of W. H. Thompson for a special case. This theorem asserts that if E = {a, b} has at most 2 elements and if c = ab-1 has odd order, then I G: (a) I and IG: (b) I must be odd. In fact, Rankin proved a more general result in which E is not required to generate G. By a cyclically ordered set I mean a sequence xl, .. ., xn of distinct elements, two such sequences (x1, ... , xn) and (Y1, .. ',Yi) being regarded as the same if they differ by a cyclic permutation, i.e., m = n and Yi = Xi+k for some fixed k (indices being taken mod n). If G is a finite group and E is a subset of G, an E-cycle in G is a cyclically ordered subset x1 ... ., xn of G such that the ratios x-1xi+l all lie in F (including x-lxl).