Date: Wed, 21 Sep 94 16:38:16 -0400 (EDT)
From: Jerry Bryan <BRYAN@wvnvm.wvnet.edu >
~~~ Subject: Re: < U, R> Group

I wanted to get back to some of the symmetry considerations associated
with the Two-Generator Group. In particular, I wanted to calculate
God's Algorithm in terms of something analogous to M-conjugates.
I haven't started the calculations, but I wanted to go ahead and
discuss what those calculations will consist of.

First, we can make the obvious observation that the <U,R> group is not
the only Two-Generator group. Any two adjacent faces can serve
as generators for a Two-Generator Group, and there are twelve such
pairs of adjacent faces. All twelve groups have identical structures,
and are isomorphic under M-conjugation.

With respect to any particular Two-Generator Group such as <U,R>, the
associated symmetry group is not M, it is a subgroup of M. Dan Hoey
has determined that M has 98 subgroups, and that the 98 subgroups
may be grouped into 33 classes. Dan has developed a taxonomy for the
33 classes and 98 subgroups. I have seen bits and pieces of Dan's
taxonomy posted to the list, and he has sent me a good bit of it
via private E-mail, but I don't think I have ever seen the whole thing
all in one piece. In any case, let's talk a little bit about the
98 subgroups and 33 classes.

Frey and Singmaster use script characters to describe whole cube
rotations. For the purposes of this note, I will use lower
case letters. For example, r will be used to describe grabbing
the right face and turning the whole cube 90 degrees clockwise
(to be distinguished from R, which means to turn only the right
face 90 degrees clockwise). In this notation, <r> = {i,r,rr,rrr}
is one of the 98 subgroups of M. (Note that <l> is the same
group as <r>.) Similarly, <u> = {i,u,uu,uuu} and
<f> = {i,f,ff,fff} are subgroups of M. The groups <r>, <u>, and
<f> have identical structures (isomorphic under rotation), and the
collection (<r>, <u>, <f>) is one of Dan's 33 classes.

Similarly, (<r2>, <u2>, <f2>) is another of Dan's 33 classes. <r2>
is a subgroup of <r>, <u2> is a subgroup of <u>, and <f2> is a
subgroup of <f>. Dan's taxonomy includes a complete description
of group-subgroup relationships within the subgroups of M, or maybe
I should say that it is a complete description of class-subclass
relationships.

The Frey-Singmaster script notation is adequate for rotations, but it
is not adequate for reflections. Instead, I will use a notation which
I believe originated with Dan Hoey (e.g., 28 Dec 1993). For example,
you could write r=(FUBD), where the upper case letters describe movements
of whole faces (*not* quarter-turns in this context!). (FUBD) means Front
goes to Up goes to Back goes to Down goes to Front, which is what happens
when you perform r. In the same notation, a Front-Back reflection
would be (FB), etc.

With that all said, the symmetry group for <U,R> is

<(UR)(DL),(FB)> = {I, (FB), (UR)(DL), (FB)(UR)(DL)}

In Dan's taxonomy, this group is a member of the W class, and there
are six such groups -- W1 through W6. I am not sure which one this one
is (I only have a list of the classes, not of the groups), but let's
say for the sake of the argument it is W3. Then for <U,R>, I will
be calculating W3-conjugate classes of the form {w'Xw} for all w in
W3. The size of the problem will be reduced by about four times,
compared to a reduction of about forty-eight times for whole cube
problems where M-conjugation can be used.

I was initially surprised that there are twelve groups M-conjugate with
<U,R>, but only six corresponding symmetry groups in M. This arises,
for example, because the <U,R> and <D,L> (diagonally opposed) groups share
the same symmetry group. I really shouldn't have been surprised.
After all, we know how many subgroups of G there are, namely
"over three beelion" (Dan Hoey, 20 Aug 1992), and we know how
many subgroups of M there are, namely 98. Since "over three beelion"
is a lot more than 98, there must be many, many subgroups of G
which share the same symmetry properties in the sense of sharing
a subgroup of M.

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
Robert G. Bryan (Jerry Bryan)              (304) 293-5192
Associate Director, WVNET                  (304) 293-5540 fax