[r-t] Alternating groups in change ringing

[Joe Norton]

On 16/04/2014 15:15, Richard Smith wrote:
> 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.

Thank you for your comments on this subject. I have been thinking hard 
about what you said which has lead me to many questions and comments. 
There is far too much swimming around in my head, so I will deal with 
things one at a time.

The first thing is something which arose from the comment that a 60 of 
Grandsire Doubles with only bobs is isomorphic A5, just like the plain 
course of Stedman Doubles. It occurred to me that if you have a course 
of Stedman Doubles with a single at the end followed by Grandsire 
pbpbpg, where g is the standard Grandsire bingle (3.145) then the 
Stedman would contain all the in course rows and the Grandsire would 
contain all the out of course rows and so must be a true extent of 
Stedman and Grandsire.

I'm sure that this has been rung many times in the past, as it is very 
simple in structure, but I am surprised that I have never heard of 
spliced Grandsire and Stedman Doubles. Or is it that I've missed 
something and it wouldn't work?!

Joe.