[r-t] Alternating groups in change ringing

[Richard Smith]

Joe Norton wrote:

> Many years ago I read a paper which I think was called "The Mathematics of 
> Change Ringing".

I don't think I've encountered this paper.  I don't suppose 
you recall the author's name?

> If I remember correctly, it talked about how sets of changes 
> form groups under row multiplication (or composition of rows, I forget the 
> exact terminology).

Sets *rows* are groups under row multiplication.  Sets of 
changes *generate* groups under row multiplication.  That 
may seem a pedantic point to make, but the difference 
between a row (e.g. 321546) and a change (e.g. the 
transpostion represented by a 36 place notation) is 
important.

(Okay, certain sets of changes are groups too if the null 
change is included: e.g. on four bells the changes {x, 12, 
34, 1234} form a group.)


> Also, I think there was talk of Cayley Colour Graphs with 
> changes forming the vertices.

The normal application of a Cayley graph to ringing has rows 
forming the vertices.  The edges represent a particular 
choice of group generators: often a set of changes, but 
perhaps also the transposition effected by plain, bobbed 
or singled leads of a method.

Two vertices, a and b, are connected iff the transposition, 
c, defined such that ac = b, is one of the group generators. 
Such edges are directed, so we should properly speak of a 
directed graph or digraph; though in the case that c^2 = 1 
for each generator c (which is the case when we are 
considering changes as the generators) the graph is 
undirected.

To take a trivial example, we know that the changes 1 and 3 
generate the extent on three bells, because we know that 
plain hunt is the extent.  We're going to label each edge on 
the graph by whether it's produced by a 1 or a 3: let's do 
this by colouring 1 edges red and 3 edges blue.  Start with 
some row, say rounds (123).  Apply the 1 change and you get 
to 132, so draw a red edge between 123 and 132.  Apply the 3 
change to rounds and get 213, so draw a blue edge between 
123 and 213.  Repeat for each of the other vertices. 
Eventually you'll have drawn the complete Cayley graph.  In 
this case it is a hexagon with alternating red and blue 
edges.

A slightly more complicated example would be the extent on 
four bells, which can be generated by the 14, 12 and 34 
changes (amongst other choices).  You'll end up with 
something that looks like a truncated octahedron.  (It's a 
bit misleading to talk about such graphs as polyhedra. 
Polyhedra are defined by their faces, and graphs don't have 
faces: just verices and edges.  You can sometimes use faces 
to represent Q-sets, but it starts to fall apart once you 
have more than three generators, or if the generators are 
not involutions.)

> More importantly, I am interested in alternating groups in 
> the context of change ringing. It is often said that the 
> dihedral group, Dn, is isomorphic to plain hunt on n bells 
> and that the symmetric group Sn is isomorphic to the 
> extent on n bells. I've always been interested in the 
> alternating group, An, which is the group of all even 
> permutations and would be isomorphic to the set of all 
> in-course changes on n bells. It occurred to me that the 
> plain course of Stedman Doubles is isomorphic to the group 
> A5, since Stedman Doubles contains only the place notation 
> 1,3 and 5 which all represent two pairs of bells swapping 
> between any two rows and hence would preserve the parity 
> of the row.

Likewise a 60 of Grandsire Doubles (called pbx3) for the 
same reason.  The lead heads and lead ends of a (standard, 
bobs-only) extent of surprise minor also form A_5.  But 
there's a rather more interesting use of A_5.  The group 
that ringers usually call Hudson's group is isomorphic to 
A_5.  Hudson's group is a transitive group on six bells 
which can be generated by the changes {12, 16, 34} on six 
bells.  Obviously it's not conjugate to the normal 
representation of A_5 (as the in-course half-extent): 
instead it's its dual under the exception automorphism of 
S_6.  Although the group has found a number of novelty uses 
on six, by far its most important use is as the part-end 
group for twin-bob peal compositions of Stedman Triples.

> I was wondering if anyone knew if much had been published 
> on alternating groups in the context of change ringing? I 
> wanted to do some investigation into the subject but I 
> don't want to reinvent the wheel if there is already stuff 
> out there that has been published on the subject!

I doubt you'll find much published on its use, for the 
simple reason that there isn't all that much published on 
group theory and ringing at all.  But the archives of this 
mailing list might be a good place to start.  Otherwise, 
ask questions.

> If anyone could point me in the direction of any 
> literature on the subject of group theory in change 
> ringing, I would be very grateful.

Brian Price's paper, 'The Composition of Peals in Parts' is 
an important paper, though more for his enumeration of 
groups up to conjugation.

   http://www.ringing.info/bdp/peals-in-parts/parts-0.html

Arthur White wrote a passable introductory paper 'Fabian 
Stedman: The First Group Theorist?':

   http://www.ringing.info/bdp/peals-in-parts/parts-0.html

RAS