[GAP Forum] Retrieving group name (fwd)

Robert Eckert eckert at math.wayne.edu
Thu Jul 1 16:55:49 BST 2004

---------- Forwarded message ----------
Date: Thu, 1 Jul 2004 01:25:40 -0400 (EDT)
From: Robert Eckert <eckert at turing.math.wayne.edu>
To: Stefan Kohl <kohl at mathematik.uni-stuttgart.de>
Cc: GAP Support <support at gap-system.org>
Subject: Re: [GAP Forum] Retrieving group name

On Wed, 30 Jun 2004, Stefan Kohl wrote:

> Robert Eckert wrote:
> > This is probably more than you want to know, but...
> >  For some time I have worked on what a "canonical" naming scheme for a
> >  Classification of Finite *non*Simple Groups would have to look like.
> [ ... ]
> This looks like a larger project --
> Some questions however (maybe the answers are somewhere hidden in your text):
> 1. Is there a bijection between your names and the isomorphism types
>    of finite groups?
That's the plan, at least:  a symbol-string should stand for either an
impossibility (there exists Z_3 X| Z_2 = D_6 = S_3 but no Z_2 X| Z_3) or a
group determined up to isomorphism; and each group (finite at least:
infinites have "self-similar" decompositions like Z = Z * Z_2 or "free
product" decompositions and maybe phenomena for which this whole language
is not suitable) should have a unique "canonical" symbol-string naming.
Problems divide into "ambiguities" (same symbol string for non-isomorphic
groups, from two series of refinements of the same Type at each step down
to the same atomics but with alpha/gamma choices that cause different
effects) and the much less serious "ambivalencies" (as between
 Z_6 = Z_2 + Z_3, Z_6 = Z_3 + Z_2,
who really cares?  any arbitrary tie-breaker rule will do).  Ambiguities
about the choice of gamma I know how to deal with; ambiguous semidirect
products are more confusing and this remains work in progress.
> 2. If not: for which types of groups or orders of groups are your names
>    unique, or which invariants do they determine?
"All groups of finite order" is the desired answer.  With certain
"honorary atomics" like Z, Q, R we can also deal with infinites that are
"solvable over the finites" (has a subnormal series of inclusions of
subgroups, i.e. each normal in the next if not necessarily in the whole,
where each quotient is abelian OR finite).
> 3. Is it algorithmically feasible to determine the name of a given group
>    according to your naming scheme? If so, for how large / complicate groups?
Very much work in progress.  First approx:  determine whether the group
has a center by taking commutator-derivation until it goes trivial or
stops shrinking; if so, try to pull out the center as a direct or ordinary
left factor:  if ordinary, keep decomposing and note whether carry is revealed
to be incompletely ordinary at some stage-- there are systematic moves
that can be made to rectify the decomposition if this happens.  But if
there is no center, use the first commutator-derived (not the terminal
stage) as your attempted H to get a semidirect by a cyclic action, or by
an abelian group of non-interacting actions, and keep decomposing to see
if one of the actions "does something funny"-- here is where I am groping
even to pin down what exactly I mean, let alone what to do about it.  The
actions should fall into inner/central-fixing cases rectifiable to direct
or ordinary, and wreath/non-wreath actions, plus only a handful of
exceptionals that need to be tagged "reverse".  The conjecture is that
the exceptionals only happen with a few "smallish" groups like A_6:  that
might prove false, but I doubt it because it is those exceptionals and the
anomalies surrounding them that allow the sporadic simples to arise; so if
there was a large counterexample, there ought to be a whole new family of
large sporadics also, and while the Simple Classification Theorem is now
scattered among hundreds of papers that I doubt any one person has read
all through, the community seems satisfied that it is correct.  A more
likely kind of failure for my scheme:  what I think is non-exceptional,
the wreath/non-wreath, conceals a kind of ambiguity (more than one
"wreathlike" or more than one "non") that I haven't considered.

To show a small example, the wreath
  (Z_3 + Z_3) S Z_2
would proceed algorithmically through the "ordinary" fork; writing the
elements 0/1/2, 0/1/2, blank/S, the action of S fixes the central {00, 11,
22} so I would try that as H and find the quotient is a D_6 = Z_3 X| Z_2
with, say, {00, 01, 02} as the Z_3 (no carry), and either 01S, which
carries by
  01S 01S = 01 10 S^2 = 11
or 00S, which does not carry but induces carry, as the non-trivial in Z_2.
This is incompletely ordinary, so it would be kicked over to the
semidirect fork of the algorithm:  the rectification is to combine the old
H with the part of T that failed to carry at all to make our new H, the
elementary-9, and treat the part of T that either carried or induced carry
as the subgroup acting on it.  Discover "carry-up" and tag it "wreath".

 For cases with a center or a smaller commutator-derived, having chosen a
tentative H, the T-elements are chosen preferring *independent* elements
(no power except id falling into H or into the cosets of H using those
T-elements already chosen) of maximal order, or else the *maximally
independent* (largest power before it falls in), and if t is chosen,
also incorporate into T the powers of t as far as independent; this
trivializes gamma so far as possible.  But for other cases, simple or
near-simple groups where commutator-derivation does not shrink the group
at all, take a maximal H: start with an element of maximal order and keep
appending elements of maximal independence that do not generate the full G
until you cannot anymore; the T that is appropriate here is constructed by
different principles, preferring involutions where possible and
incorporating conjugates rather than powers; if the group is not actually
simple, decompose as if it were the simple group anyway, and the extra
factor will have to be detected.
> Depending on the answers to these questions I think it might be *very* desirable
> to have these names and a corresponding group identification routine available
> in GAP, and I would recommend that you discuss this with the author of the
> small groups library, Bettina Eick.
That might be premature at this time; on the other hand, maybe working on
an implementation would clarify my mind about what the "rules" really need
to be.
> Another question: How do you suppose the `Affine Monster' to look like,
Like a warthog who can't get a date even from female warthogs.

> and with which kind of methods do you try to establish its existence,
> i.e. do you use a constructive approach (construct a permutation representation,
> matrix representation, finite presentation, ...) or a non-constructive one?
Gross framework of the "big lemma":  work in permutation-representation
terms.  Given an arbitrary finite simple G, choose a maximal subgroup H
and action on H-cosets is permutational of least possible degree-- unless
there happens to be a different maximal subgroup of smaller index
reachable by a different route?  H is a one-point stabilizer: label the
point standing for H-itself as "point infinity".  Pick a "point 0" and an
involution t_0 in that coset; now find a "wholly regular" element of H,
meaning one that has no power short of id that fixes any point but
"infinity" (no mixing of different lengths in the disjoint-cycle
structure), and build up to the largest possible wholly-regular subgroup
(at least containing that cyclic, but as much as you can add and have no
non-trivial element acting less than freely on the points other than
infinity)-- unless there isn't even one regular in H that doesn't mix
cycle lengths?  Start the rectified T as t_0 (moves "infinity" to "0")
plus its conjugates by the big wholly-regular subgroup (each moving
"infinity" to a different point); if that reaches all the points we are
done (as in PSL(2,q) but not in general); else find an independent wholly
regular element of H moving "0" to one of the points not reached by the
orbit of the earlier subgroup-- unless that doesn't exist?  Else continue
adding blocs of conjugates to the transversal until complete.  Regardless
of what the simple group, you should be able to rectify thus far.
If, instead of an arbitrary simple group we are given one equipped with a
canonical decomposition, the rightmost pseudo-group factor will be
defining S, the "simple core"-- unless all H is solvable?  The
factors right of that pseudo-group will contain some multi-point
stabilizing stuff (the "multiplicatives") and what we want to be
the initial big wholly-regular subgroup (the "additives")-- unless it just
isn't there?  (For example, in PSL(n,q) the H we have in mind is the
subgroup of matrices with the bottom row all zeroes except the rightmost
entry; this breaks down as PSL(n-1,q) which is our S, one new
multiplicative group, and a column of new additive groups which t_0 is to
be conjugated by to get us started off).  Then the further regular
subgroups that suffice to fill out T should be precisely the same ones
which sufficed to build the pseudogroup for S.  If everything goes
through, we get diagrammatic relationships and can show how to figure out
which diagram it is.

The question marks are the places where things could fail, and that should
mean G isn't really simple, or H really isn't maximal, or S wasn't
diagrammatic either, or it can mean a different H should be chosen-- but
you can get two H's pointing their fingers at each other saying "use that
one instead"; this comes up in the "set-product" case.  I think I can get
to "if everything was diagrammatic and typical before, it will be again".
The problem is that I also seem to be finding indications of the converse,
"if the simple core is weird, there is going to be a weird way of carrying
it on."  In the alternating groups, this is true:  every one is a base for
building the next one.  I don't want to prove that for every sporadic
group there exists a next-larger one, because it doesn't seem to be true,
and I find it embarrassing to prove falsehoods.  Given a simple core, if
you are then going to append some intervening factors and a capping
pseudo-group that closes up as a new simple group, the nature of the
pseudo-group defining the simple core puts constraints on what this
continuation has to be; the diagrammatic case imposes some complicated
constraints (intervening additive and multiplicative factors that tie
back just so); the alternating groups are certainly unlike most sporadics
in demanding a very simple constraint, no intervening factors period, old
simple core maximal in the next simple group; but each sporadic wants
thus-and-so, blah-blah-blah, lah-di-da, and it is hard to see why the
constraints would boil down to a contradiction, just "can't be done",
especially hard to see why this would happen in one case only.

This is not what happens in the "terminal" diagrammatic cases, the E_8,
F_4, and G_2 exceptionals:  if you add a node you get affine diagrams, and
what that means for the E_9, F_5, and G_3 groups is that the constraints
will demand intervening factors which need some more which need some more
which need...  Let us look at some easier groups to get the picture.  I
decompose Z as ... * Z_2 * Z_2 * Z_2 (or you can use all Z_10 if you like
decimal digits better), each bit carrying to the next higher-order,
indexed 0 to infinity:  but with the restriction that an element of
Z has all bits 0 past some N, or else all bits 1 (the negatives: in base-10,
negative 1 would be written all 9's).  This I would write
asterisk-with-a-bar-over-it Z_2, the bar for "ceiling", as opposed to
asterisk-hat, for "metric completion", an infinite ordinary product of
Z_2's indexed 0 to infinity, but without the restriction:  that
construction makes the p-adics, and it is not isomorphic when you change
base anymore.  Asterisk-underlined is product of Z_2's indexed minus
infinity to 0:  this is the lesser Pru"fer group (he called it
"quasi-cyclic") of fractions whose denominators are all powers of two (not
isomorphic if you use a different prime), modulo one (no carry out of the
topmost bit, it just overflows); putting a wedge (upside-down hat) instead
of bar underneath means no restriction that you get down to a sea of
zeroes eventually, and this is R/Z, the reals from 0 to 1 (and isomorphic
if the base is changed); bar over and wedge under is R.

  Similarly you can make various infinite iterations of direct products,
like the infinite elementary-2; additive Q is a direct product of the
greater-Pru"fers (bar over and under an asterisk:  fractions, whether
greater or less than one, whose denominators are powers of p) for each
prime p; multiplicative positive Q a direct product of an infinite number
of copies of Z ("logarithms" for the p-factor at each prime p); both with
the all-but-a-finite-number-trivial restriction.  Semidirect products
would not seem to admit this kind of iteration, but what the "Affine
Chevalley" groups ought to look like is some infinite tuple of groups
joined by operators like we find in the move up from PSL(n,q) to
PSL(n+1,q), where a column of additive groups acts each on the previous
until a multiplicative group acts on them all and a pseudo-group wraps it
up-- except that there is "column" after "column" after "column" in some
rhythmic pattern, an infinite number of times "before" the pseudogroup.
It still should be possible to construct a systematic description of what
kind of infinite tuple constitutes an element of an Affine Chevalley, and
what the product of two elements is defined as-- not that I feel like
sitting down and writing out such a description right now!

So, the Monster shouldn't lead to "no continuation possible" but to "no
continuation can close in a finite number of steps", at least that is my
hunch.  Besides the Dynkin diagrams there are the H diagrams, using braid
numbers 5 and 5/2 with Dynkin edge-number D(B) = 4 cos^2 pi(1 - 1/B)
working out to phi^2 = 2.618... and 2-phi = .382... (for sound reasons
stemming from the non-integral Dynkin numbers, they support no Lie-algebra
structures and thus aren't heard of as much); these describe the fivefold
symmetric polytopes, like my "good friend" H_3 who lives in the
icosahedron.  You approach H_3 groups which act on a minimum of 12 points
through the one-point stabilizers with their peculiar actions on 11:  this
is so strongly reminiscent of the Mathieu groups that there must be a
connection, although I am damned if I can tell you what it is.  There is
an affine ~H_4, forked with a central node connected to three others by 1
(braid-number 3), .382 (5/2), and 2.618 (5) edges:  the affine nature
indicates the linear dependency between the vectors defining the
icosahedron and those of Kepler's stella minor figure (12 pentagrams
joined 5 at a vertex, using the same vertices as the icosahedron, but a
set of three which would be a triangle in the icosahedron have no edge
connections here at all).  To make these reflections act independently,
you have to change the figure into a multi-sheeted Riemann cover,
festooned with branch points so that a series of moves which ought to
bring you back home instead leaves you at the analogous place one sheet
up; this structure is maddeningly confusing, governed by some kind of
"Affine Mathieu" group that I need to get a grip on.  Then there is an
affine ~H_5, an unforked chain with 2.618, 1, 1, .382 edges, describing a
linear dependency among the 10 stellated polytopes that can be inscribed
in the large 4-dimensional hypericosahedron/hyperdodecahedron polytopes;
again there is, or should be if I could figure it out, a way of turning
this into a Riemann cover where paths that should be loops do not bring
you home.  Inside THIS labyrinth is where the "Minotaur" (my pet name for
the "Affine Monster") ought to live.  I feel that if I could just write
down the reiterative description of what the Minotaur is, it would be
immediately "obvious" not only that it has to be infinite, but that it is
the Monster's only son.
> Best wishes,
  Bob X

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