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On 14 Jan 1992, Allan C. Wechsler posted

Regarding the meta-approach of descending through a series of subgroups,

how much leverage does properly selecting the chain give you? It seems

like the jump from <F2,B2,...> to <F2,B2,L2,R2,...> is pretty large.

There are probably other paths through the subgroup lattice.Does anyone have a table of subgroups?

As far as selecting the chain goes, I have been meaning to look into

that a bit. Of course, since Bill posted, the results of Hans

Kloosterman, Michael Reid, and Dik Winter have shown that you indeed

get a lot of leverage. I would like to get some idea of the possible

group towers, for a more general approach to selecting which towers

give you leverage. But what I haven't been able to figure out is how

to figure out which coset of G1 wrt G2 you're in. I've been able to

figure it out for specific groups, but if we wanted to do this for a

lot of chains, we would need to do coset identification given G1 and

G2 as a table of strong generators. We could in fact ensure that the

strong generators of G1 form a subset of those of G2. Is that a hard

thing to do?

More to the point, I've heard that the FHL algorithm should more

properly be called Sims's algorithm and that Furst, Hopcroft, and Luks

mostly analyzed the performance. I haven't read anything by Sims on

it, though. Is there a good reference that treats this sort of

algorithm in a more general setting? I have toyed with implementing

the Jerrum improvements to FHL, but it is a mighty complicated beast.

Also, a talk announced in the archives mentioned 1987 work by Akos

Seress that was supposed to be an improvement, but I don't know

whether it got published. Anyway, if not, do you know if there is a

good general way of finding out which coset a given position is in.

On 29 Jan 1992, wft@math.canterbury.ac.nz (Bill Taylor) posted

There hasn't been any response to this, seemingly, which is a pity.

For some reason, I never saw Bill's message. I just noticed it when

comparing my files against the archives. [ Archives seekers note: the

archives have moved to FTP.LCS.MIT.EDU (18.26.0.36), still in

directory /pub/cube-lovers. ]

In any event, I would like to know of any other well-known subgroups.

There are the slice group; double-slice group; U group; square group;

anti-slice group. How many others are there not mentioned here, that

people know of ?

There were some tables in Singmaster with more examples, and there are

the stuck-faces groups that I wrote about on 21 July 1981. I seem to

recall there was some non-obvious equivalence between two groups,

perhaps the slice group and the antislice group. But a general list

of popular subgroups would be interesting. Of course a list of *all*

the subgroups would have, um, over three beelion of them. I suspect

it has more than 4.3x10^19. Does anyone know a good way of counting

how many subgroups there are? Or even a way of estimating the number?

By comparison, the symmetries of the cube form a 48-element group with

98 subgroups.

Dan Hoey

Hoey@AIC.NRL.Navy.Mil