[r-t] Changes generating groups
Alexander Holroyd
holroyd at math.ubc.ca
Wed Sep 15 19:58:43 UTC 2004
> What are the properties of a set of changes that generate a group? Is it
> obvious that <-,1,34> works?
Perhaps I should have been more precise (again!). Any set of rows
generates a group. In some cases it will be the whole extent, S_5.
In this context, I am interested in cases where the generating rows are
all legal place notations. Then it is likely possible to find a method,
using those place notations only, whose plain course is the group.
There are not so many interesting possibilities. Most sets of place
notations either generate the whole extent (E.g. <-,1,4> on 6) or
something fairly trivial involving splitting the bells into disjoint sets
(E.g. <-,2,34> is a lead of MUG minor - 8 rows). Another non-trivial
example is <-,1>=Original.
It's obvious that for example <-,2,34> will have the bells split into 3
pairs 12,34,56, since there is never any traffic between 23 or 45. I
don't know a general easy way of seeing whether a set of rows will
generate the whole extent or not. But for <-,1,34> it is obvious; all 3
p.n.s are left-right symmetric, so bells 1&6 will always be in either
16,25 or 43; and similarly for 2&5 and 3&4. A little more thought reveals
that one can get all such rows, so the group has order 3! x 2^3 = 48.
Eureka Minor seems a bit different. One lead (started from the 36) is
_half_ of the group <-,1,34>, and is not a round block. Perhaps someone
can enlighten us further about how it ws discovered?
> In the same vein, what are the fundamental (group?) properties of methods
> that enable an extent to be produced? It's not at all intuitive to me that
> Cambridge Minor, say, can produce all 720 rows in such an elegant way.
The reason is the same one Richard explained. The lead-ends and
half-leads of the extent form a group - A_5 (order 60, with the treble
fixed). The half lead of the method is a traversal of the cosets. In the
particular case of this group it's easy to see that the 12 cosets
correspond to the 6 possible positions of the treble, with + or - parity
for each. Then one can check that any standard TD method traverses
them.
For a more detailed exposition of these ideas one could start by reading
Price's "Composition of Peals in Parts":
http://www.ringing.info/bdp/peals-in-parts/parts-0.html
Ander
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