Hudson's Group and A_5

All numbers of bells use permutation groups. Compositions divided into parts are permutation groups, but methods too are underlying permutation groups. Even call-change sequences can be structured in this way.

 

Though this is so on all numbers of bells, Minor is a particularly interesting example. S_6 (the group of all 720 changes on six bells) has a unique outer automorphism. The result is that some groups in Minor have a dual correspondence with one other group (if not, they transform to themselves.) That correspondence can be with another Minor group, or - more interestingly - it can be with a Doubles or Minimus group.

 

Dual correspondence means that the elements of one group can be 'mapped' onto the dual group and vice versa. Brian Price devotes a section to this in his paper, and describes it as the dual groups 'pairing off.' So if you found just one of the groups, you would be able to identify the partner. He lists all of the dual pairs.

 

For me one of the most interesting examples is Hudson's Group and A_5. Hudson's Group exists on six bells and is most useful in Stedman and Erin Triples compositions. This is because Hudson's Group contains no single cycles of three bells at all, and as a single cycle of three bells permute on the front of a Stedman six, we cannot have part ends that include...well...single cycles of three bells. You find the group popping up in other areas of composition too, including Minor Principles, Twin-Hunt Triples, and variable-treble Minor (more of which below). One version ( or 'isomorphism') of the group can be found on this page.

 

A_5 stands for the alternating group on 5 bells. These are all of the 60 positive changes on 5 bells, and is half of the extent (half of S_5). So if you rang a plain course of Stedman Doubles or a bobs-only 60 of Grandsire Doubles, you would have rung A_5. If you rang a 720 of Cambridge Minor, the 60 lead ends and lead heads would have been A_5.

 

Hud60 and A_5 both contain 60 elements. To transform A_5 from Hud60, you need to exchange conjugacy classes. This sounds complicated, but it's actually quite simple and familiar. In the full group of S_6 (all 720 changes on six bells), there are a certain number of ways that we can break the six bells into cycles. These different types of cycle are the conjugacy classes. In S_6 the conjugacy classes are:

 

Conjugacy class        Cycle type

(i)                               (123456)

(ii)                              (123)(45)

(iii)                             (12345)

(iv)                             (1234)

(v)                              (1234)(56)

(vi)                             (123)

(vii)                            (123)(456)

(viii)                           (12)

(ix)                             (12)(34)(56)

(x)                              (12)(34)

(xi)                             (1)

 

What does this mean? Note that we aren't using transpositions like we would in ringing notation. All bells contained within a pair of brackets do a cycle.

So in (i), all of bells 1,2,3,4,5,6 do a cycle, a 6-part transposition of some kind. You may ask 'which transposition.' The answer is, it doesn't matter. It could be 241635, 462513, etc (the backstroke changes of Plain Hunt on Six). It could be 234561, 345612, etc (cyclic part ends.) In group theoretical terms, these are identical but just happen to look different (they are 'isomorphic' to each other.)

 

In (ii), bells 1,2,3 do a cycle, whilst bells 4,5 keep swapping over. Bell 6 does not to be written down, because it is not involved in any of the maneuvers. Again, bells 1,2,3 and 4,5 are only an example. It could be bells 2,4,5 and 1,3. It applies to every partitioning of bells. 

 

As I said above, dual groups are produced by exchanging one of these conjugacy classes for another. A_5 is on five bells, so it only contains (ii), (iv), (vi), (vii), (x) and (xi) from the list. A_5 transforms to Hudson's Group by exchanging (vi) for (vii). So any 3-bell cycle in A_5 - such as (123), (526), (236), etc - is exchanged for paired cycles of three bells: (123)(456), (526)(134), (236)(145), etc.

 

This is why Hudson's Group doesn't contain any single cycles of 3 bells. Exchanging conjugacy classes in this way will find all of the dual groups that are shown in Price's paper.

For the record, the other exchangeable conjugacy classes are: (i) and (ii); (viii) and (ix). The other ones remain preserved (and are thus 'self-transforming') 

 

Examples in composition

The isomorphism of A_5 and Hud60 isn't usually obvious in actual compositions. This is because the internal falseness of particular methods usually camouflages the fact, making us think that there can be no possible relation between the two groups. For example, A_5 is useless for peals of Stedman Triples. The part ends are all 60 positive changes on the front five bells, with bells 6,7 as observation bells. But these 60 part ends include cycles of 3 bells - like 12345, 31245, 23145. Whenever bells 1,2,3 meet each other on the front in a six, this six will run false against the equivalent sixes in other parts. But Hud60 is absolutely ideal for peals of Stedman Triples, because it doesn't contain any of these 3 bell cycles. As we saw, they were exchanged for 'bicycle' transpositions, meaning that part ends 123456, 312456, 231456 become 123456, 312645, 231564.

 

However, there are a few rare examples of when the dual correspondence between A_5 and Hud60 can be seen in composition.  

 

Recently I produced some 60-part peals in twin-hunt Triples methods, using Hud60 part ends. It was an interesting idea; whilst in the garden it suddenly occurred to me for the first time after all these years that Hudson's Group could be used for Plain Triples as there are enough inside bells (just.) More information about the construction of those peals can be found elsewhere. The peal of Grandsire made up from the P-Blocks is strikingly similar to an old peal by James Lockwood. Essentially, the peals only differ by one call in the written notation. Yet Lockwood's peal uses A_5 part ends, compared to my own Hud60 peal. This is plain to see, as Lockwood's peal has a fixed bell (the 7th) in every part (remember, every part is only 6 leads long), compared to mine which has no fixed bell at all. The largest subgroup common to A_5 and Hud60 is the dihedral group D_5 (10 elements.) These are the 10 'main' part ends in both peals. So the fact that the sub part ends in both peals are D_5 helps to make the peals look even more similar and highlight the relation.

 

This leads nicely onto variable-treble Minor compositions.

 

In about 2004 it was discovered (or re-discovered) that some Minor methods traverse all the cosets of Hud60 in a half lead. In other words, methods like Hudson Delight Minor (now known as Norwich Delight) can be rung as a 60-part 720, each half lead being a part. Special singles alter the main hunt bell. But you can still call W,H,W x3 for a standard, treble unaffected 720. The first way is Hud60, the second way is A_5 (where each lead end and lead head is a part end.) The underlying thread of D_5 is a course of the method (where each lead end and lead head is a part end from a D_5 group.) 

 

The usual 720 for variable-treble Hud60 parts is to call two singles a whole lead apart. This shunts the bells into a 5-part transposition, whilst sweeping up the 6 leads needed for 1/5 of a 720. Nothing else needs to be done: the composition is finished! I came up with an Mx method (i.e: same lead end order as Kent TB) where the usual 5-part 720 doesn't work. Instead I came up with a 3-part:

 

 123456  2 5

 543216  x x
 231645  x

 3 part

 x = 34 Extreme.

 

Now this is very similar to W,H,W x3. The call string is identical to W,H,W x3 in Bourne Surprise Minor. In fact, it is fair to say that the two 720s are parallel.

Both are 3-parts, both have three calls in each part, a call causes a three-part transposition, and an irregular q-set is used to join an even number of courses together and shunt the bells into a part end order. As explained above, Hud60 transforms from A_5 by exchanging (123) with (123)(456), and that couldn't be clearer in this example.

 

When I wrote about this on the ringing theory email group, Andrew Johnson provided another very interesting example of the A_5/Hud60 relation being evident in composition. The first peal of Stedman Triples he gives is remarkable for its similarity to W,H,W x3.

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