[r-t] Re: Proofs

[Richard Smith]

R.P.I. Lewis wrote:

> > X(AC) here means just the falseness between A & C - no 'direction' is
> > intended but having decided on the rule 'inverse of second of the pair
> > operating on the first of the pair', we're stuck with it.
>
> if there was really no direction then why do we not get X(AC)=X(CA)? Seems
> to me that the fact that X is not symmetric tells us something

This is partly because it is not the individual X(A,C)s that
are of interest -- it is the set, F(M), of X(A,C)s for each
A and C in a lead or course, M.  This set does have more
symmetrical properties; for instance,

  f in F(M)  <==>  f^-1 in F(M).

> as in "J=12"? Is this a standard notation, and if so where should I have
> looked to find this out for myself?

Robin is taking advantage of the symmetries of the whole
course.  In particular, he is assuming that the method is
palindromic, and either that the method has regular (i.e.
Plain Bob) lead heads, or that it has a seconds place lead
head (either are sufficient).   For simplicity lets assume
the former.

Now, if we write D as the set of regular lead heads and ends
(which form the group generated by the place notations 12
and 18 and which is isomorphic to D_7), we can write the set
of rows in a course, C, in terms of the set of rows half a
lead, M:

  C = MD

where the MD is just a notational convenience to mean each
element of M multiplied by each element of D.  If we extend
this notation a bit further, we can write the table of false
course heads, F(C), as

  F(C) = C^-1.C

wherre C^-1 is the set containing the inverses of each
element of C.  (This is just Robin's X(A,B) but letting A
and B run over the whole course.)

A bit of rearranging gives

  F(C) = D M^-1 M D
       = D F(M) D.

That is, the table of false course heads, F(C), is entirely
fixed from the table of false lead heads, F(M).  Coupled
with the earlier observation that

  f in F(M)  <==>  f^-1 in F(M),

this means that given

  f in F(M)  ==>  D {f, f^-1} D in F(C).

This partitions the space of lead heads into the 28
named fixed-treble "falseness groups".  (If you include
non-fixed-treble course heads, you get 143 falseness
groups.  These are rather less useful, though.)

> I
> *thought* that what you were doing was to take the values of X and
> generate a subgroup, (which would hopefully tell us something about the
> method)

Yeah.  That can be a useful technique too.

Richard