I have known for many years, in principle, that group theory is relevant to the art of change-ringing of bells. But when I first came across this idea at the age of 16, I did not understand the concept of a coset. Without understanding cosets, it is impossible to understand how group theory is used in bell-ringing. By the time I understood cosets, a year or two later, I had lost interest in bell-ringing.

Fast forward 45 years. A chance question from a bell-ringer about the history of groups, a week or so ago, re-kindled this lost interest. It became clear that in the translation from pure group theory to bell-ringing I had confused the elements of the group with the cosets of the group, and this confusion had persisted in my brain for 45 years without any apparent intervention from outside. Faced with the experimental evidence, however, I was forced to confront this confusion and deal with it. Of course, I am not confused about the difference between elements and cosets in abstract groups – that would be absurd. I was confused only about how these concepts map to the appropriate concepts in the application. I wouldn’t say that I now understand the mathematics of change-ringing in its entirety, but I understand the principles on which it is based. After a few days of practice, I have a repertoire of techniques that I can use to compose peals of varying lengths on various numbers of bells. Not that I can compete with hundreds of years of accumulated experience by the expert bell-ringers and composers, of course.

Two points I would like to make. First, that bell-ringers were using groups two centuries before they were defined by Galois. Admittedly these were mostly dihedral groups and symmetric groups. But, secondly, in 1814, I am told (and I have no reason to doubt this), Hudson discovered what we now call PSL(2,5) permuting six bells, and used it for composing peals. Two decades before Galois.

Now it gets interesting for group theorists. What happens for PSL(3,2) on 7 bells? Known in bell-ringing circles since 1906. Why 1906? PSL(3,2) was known to Galois, and featured in Camille Jordan’s 1870 book, but did not become generally accessible until L. E. Dickson’s book in 1901. Is this relevant? Perhaps there is a PhD for a historian of mathematics in this story. And what about PSL(2,7) on 8 bells? Before anyone gets started on the idea of using M11 on 11 bells, let me point out that you have to use it for a full peal of 11! changes, which would take you a couple of years of non-stop ringing, or at best a half-peal of 11!/2 changes. Think what might happen if someone made a mistake and you had to start again… Of course, M11 was known in the 1860s, so it is not impossible that someone might have thought of this, before discarding it as obviously ridiculous.

Now it gets interesting for sociologists. What has been the reaction of bellringers to my quite naive and ill-informed attempts to develop the group theory they use? They reply to my emails. They suggest problems to look at. They appraise my compositions and make polite remarks about them. They forward my emails to their friends. What has been the reaction of (theoretical) physicists to my much less naive and much less ill-informed attempts to develop the group theory they use? The exact opposite.

Group theory in both bell-ringing and physics is quite straightforward, once a couple of initial hurdles are overcome. A professional group theorist can help in both endeavours. In change-ringing the important issues are well-understood, and the techniques have been well-developed over centuries, so the group-theorist’s potential influence is quite limited, but not negligible. The theories are well-tested in experiments, so there is no doubt about the exact correspondence between the two. In physics, the use of group theory goes back only one century, is not well understood, and has changed only over a time-span of about 50 years, up to the 1970s, after which no development at all has taken place. There are significant ways in which the theory does not agree with the experiments. The group-theorist’s potential influence is therefore enormous, if only the extraordinarily arrogant and dismissive attitude of theoretical physicists could change a little bit.

June 20, 2020 at 4:29 pm |

I was being generous to theoretical physics. The applications of group theory in fundamental physics developed once in the 1920s, once in the 1960s, and once in the 1970s. That is all.

June 22, 2020 at 5:55 pm |

The uses that bell-ringers put groups to are not necessarily exactly the ones that group-theorists naturally think of, which is what makes this application interesting. Bell-ringing compositions are not always groups, but are often set products of subgroups of the symmetric group. For a group theorist, therefore, there are some particularly interesting cases when the symmetric or alternating group is a product of two disjoint subgroups.Two cases on even numbers of points that seem to be well-known to bellringers are S_6 as a product of PGL(2,5) and S_3, and S_8 as a product of PGL(2,7) and S_5. These are in addition to the “obvious” factorisations on odd numbers of points, for example of S_7 as a product of D_{14} and A_6. Since 8 is a common number of bells in a belfry, the factorisation PGL(2,7).S_5 is a rich source of compositions. A full peal takes the best part of a day, but by taking subgroups of PGL(2,7) and/or S_5 one can obtain all manner of performances of almost any length. Since the group generators are constrained to be involutions swapping (any number of) pairs of adjacent bells (in temporal order, not spatial order), the basic building blocks are dihedral groups. In PGL(2,7) one can choose a dihedral group of order 16, 14 or 12, and in S_5 one can choose 12, 10 or 8. All 9 combinations can be performed, some in multiple ways. It is not at all obvious that PGL(2,7) can be performed on 8 bells, but Isis Major is one such composition, first performed in 1986. A composition for PGL(2,5) on 6 bells is called Striking Minor, which I suppose is meant as a pun (Minor=6 bells, Major=8 bells).

June 25, 2020 at 9:04 pm |

No-one could give me a composition for PSL(2,5) on 6 bells, so I made one for myself. I doubt if it is really new, because the group theory implies that the number of choices is really very small. I assume that Hudson in 1814 found something similar to what I found, but I could be wrong. PGL(2,7) on 8 bells is really hard – at least I was unable to find anything new, and as far as I know, Isis Major may be the only one known. it seems to me that PSL(2,5) is the only simple group, other than alternating groups, that can actually be rung. I looked at PGL(2,q) in general, which also seem to be impossible. At this stage it seems to me that the only contribution that a group theorist can make is (a) to examine factorisations of groups, and more generally, pairs of subgroups each of which has trivial intersection with all conjugates of the other, in order to suggest new “methods” of ringing, and (b) to show that other proposed methods of various types cannot work. The actual art of composing methods for change-ringing involves far more than group theory, including combinatorics and of course aesthetics, and is not something that I feel qualified to pursue.

The contribution of group theory to physics is of much the same type. It does not reach into the technical details of the applications in any way. But it can analyse what methods might work, and what methods definitely cannot work. I can prove, without a shadow of a doubt, that the group theory that is currently in the standard model of particle physics cannot work as part of any unified theory, in any way whatsoever. This is pure group theory, and is completely independent of the details of the applications to which this group theory is put. On the other hand, I can suggest group theoretical foundations that might work (in the sense that I cannot prove that they cannot work).

To illustrate the point, there are many compositions of 168 changes of 7 bells. It is natural to assume (as I did to begin with) that the simple group of order 168 is involved. It is not. Such compositions often divide into 12 cosets of a group of order 14, or 14 cosets of a group of order 12, and these groups are often dihedral groups. Someone who is not very experienced in group theory might assume that there is a group of order 168 involved, namely the direct product of a dihedral group of order 14 and a dihedral group of order 12. There is no such group involved. The symmetric group S_7 does not contain such a subgroup. Nevertheless, the 168 changes exhibit both dihedral groups of symmetries. But they combine in a much more complicated way.

The same is true in physics. The weak force exhibits an SU(2) symmetry, and the strong force exhibits an SU(3) symmetry. It is natural to assume that there is a direct product of SU(2) and SU(3) describing the symmetries of physics. There is not. These two groups (or something like them) embed in a larger group in a much more complicated way than a direct product. Of course, I cannot explain all the detailed physics that goes on top, just as I cannot explain all the details of change-ringing. But I can explain the group-theoretical foundations in both cases. Groups are extremely unforgiving structures. They are extremely rigid, and what they say cannot be contradicted, not by bell-ringers, and not even by physicists. The sooner that theoretical physicists learn this, the better.