[r-t] Alternating groups in change ringing
Joe Norton
strainsteamford0123456789+ringingtheory at gmail.com
Thu Sep 18 13:27:55 UTC 2014
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.
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