[r-t] Alternating groups in change ringing
Richard Smith
richard at ex-parrot.com
Wed Apr 16 14:15:59 UTC 2014
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
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