[r-t] Hudson's Group and A_5

[Richard Pullin]

Recently I produced some 60-part extents of twin hunt Triples methods,
using Hudson's group 'Hud60' as part ends. These can be found here:
https://complib.org/collection/10788

The peals are easy to learn but disadvantaged by lots of in-course singles
and no fixed observation bell, so they are probably more of theoretical
interest.

Two P-Block peals - one of Grandsire, one of Single Oxford - use just one
kind of call, and can of course be transposed for any 7ths or 5ths place
method.

The Grandsire peal is extremely similar to an old one by James Lockwood,
both in the link above. Lockwood's peal uses the alternating group A_5 as
part ends, i.e: all 60 +ve part ends with one bell fixed (the 7th in his
peal.) As my peal uses Hud60 as part ends ('a part' still meaning every
6-lead course) there is no fixed observation bell, yet both peals are
almost identical except for one call in the written notation.

As has been discussed on this list and in a Brian Price paper, there is a
dual correspondence between A_5 and Hud60, due to the unique outer
automorphism of S_6. The rotation symmetries of an Icosahedron can produce
A_5 and Hud60, a description of how to do so being posted on this list in
April 2008. Are the two peals by Lockwood and me are a rare example of this
correspondence being obvious in a ringing context? Usually internal
falseness gets in the way, like the Stedman sixes working against A_5 part
ends yet still being so ideal for Hud60. But in the P-Blocks of twin hunt
methods there is no in-course falseness at all, so no favouritism towards
either group.

Other examples of A_5 meeting Hud60 in ringing composition. Hudson Delight
Minor (aka Norwich Delight) has been discussed on this list - a half lead
of the method traverses the cosets of Hud60, thereby making variable treble
720s possible based on this group. But you can still call W,H,W to make an
A_5 extent, with the unaffected treble being the ' fixed observation bell.'

For a similarly constructed method that has an Mx lead head, you can call
PXPPXPXPPP thee times (231645 first part end) where X is 34 at the lead
head. The call string for this 720 is identical to W,H,W in Bourne S Minor,
and a closer inspection reveals that both 720s are virtually parallel.

The biggest subgroup common to Hud60 and A_5 is the dihedral group D_5
(order 10). From this vantage point it is interesting to see how A_5 and
Hud60 diverge. For example, all palindromic Minor methods have the 10
elements of D_5 as their lead heads/ends. The outer automorphism of S_6 is
apparently based on exchanging conjugacy classes between the dual groups.
So if we switch the 3-bell cycles like (123) in A_5 with cycles (123)(456),
we get Hud60. Lo and behold, the parallel 720s just described are so
similar to each other because the underlying thread of D_5 is used in both
cases as the method, but the difference is that we get two trios of bells
(231)(645) rotated instead of one 1(342)56 at the part ends. .

Some of my B-Block Triples 60-parts could also be rung to A_5 part ends
with very little adjustment. The peals of Single Oxford, Double Oxford and
Single Court all have a fixed bell in the 6-lead part which rings in every
lead position once, as the 7th does in Lockwood's peal.

The St Clement's peal cannot be rung to A_5, as no one bell traverses all 6
lead positions within the part. The same is true of the Double Grandsire,
making both peals 'genuine' Hudson 60-parts. In the St Clement's case this
is not surprising, and it was the very first method I tried for a Hudson
60-part. The important feature of Hud60 is that it doesn't contain any
cycles of 3-bells whatsoever, which is why it is so useful for Stedman and
Erin Triples, where 3 bells permute in every six. As St Clement's Triples
is like an 'Mx' method, with a single trio of bells cycling round in 2-3-4
at the bobbed leads, this makes the B-Blocks ideal for Hud60: you basically
have the extent handed to you on a plate.

For the sake of completeness I wanted to produce a Hudson 60-part in a
single hunt method like Plain Bob or St Simon's. Though I managed to come
up with some 6-lead parts I could never get the bells into anything other
than a 2-part shunt, which isn't much good for making structured peals. The
in-course falseness of these methods works against the group. (For peals
based on A_5, the falseness can be avoided by calling sB,sH in every
course.) I was equally unsuccessful with 3-lead course methods like Reading
Old Bob. It's not often that twin hunt methods are easier to compose for
than single hunt!
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