[GAP Forum] Retrieving group name
Robert Eckert
eckert at math.wayne.edu
Wed Jun 30 15:47:01 BST 2004
On Tue, 29 Jun 2004, Stefan Kohl wrote:
> Igor Schein wrote:
>
> > I am struggling to find the way to retrieve canonical group names
...
> However there are no canonical names for larger groups --
> ...Note in this context that distinguishing groups of orders with
> `many' equal prime factors is often quite difficult, and that one
> would need a very elaborate and complicate naming scheme for them
> if one would like to have descriptive names.
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 is the language I use: an *alphabet* for a group G is a nested
inclusion of subgroups {1} = H_0 in H_1 in H_2 ... in H_n = G, with right
coset transversals T_i, 1<=i<=n so that H_i = H_i-1 T_i. Each group
element can thus be written "Chinese restaurant menu" style (one from
column A, one from column B...) as t_1 t_2 ... t_n and calculations
performed if the *combing* rules for rewriting t' t where t' in T_j but t
in T_i, i<j, into a string of t-elements in proper order are all known.
A *refinement* inserts one more H, breaking down one T; an *atomic*
alphabet admits no further refinement, and a group is named by choosing a
standard series of refinements down to atomic level, using an operator
symbol to indicate the type of refinement done at each step and
parentheses to indicate the order of these steps. For example, A_4 would
decompose with a Klein-4 subgroup as H_1 and a cyclic-3 as its T_2, then
decomposing the Klein-4 into cyclic-2's, thus
A_4 = (Z_2 + Z_2) X| Z_3
where + (circled-plus where available) marks a direct-product and X| a
semidirect.
The decomposition types are coded by the maps used in the combing of
(h_1 t_1) (h_2 t_2) into proper order:
h_1 (t_1 h_2) t_2 = h_1 (h_a t_b) t_2 = h_1 h_a (h_c t_p)
Define alpha(t_1) : h_2 -> h_a as the *action* of t_1 on H; t_1 *acts
trivially* if alpha(t_1) is the identity map, and *alpha is trivial* if
all T acts trivially.
Define beta(h_2) : t_1 -> t_b as the *back-action* of h_2 on T; h_2
*reacts trivially* if beta(h_2) is the identity map, and *beta is
trivial* if all H reacts trivially.
Define gamma(t_b, t_2) = h_c as the *carry* T x T -> H; *gamma is
trivial* if this is always the identity element. For any t in T let
t-bar be the coset representative of t^-1, then gamma(t, t-bar) is the
*internal carry* of t.
Define pi(t_b, t_2) = t_p as the *factor product* in T, not the same as
the group product in G if gamma non-trivial, and not even associative if
Image(gamma) reacts non-trivially. At a minimum this product has an
identity (the coset representative of H itself, for which there can be no
reason to choose anything except the identity element of G) and inverses
(although in perverse cases of non-associativity, the left-side and
right-side inverts might not be the same).
H is normal iff beta trivial. T is a subgroup of G iff gamma trivial. T
is self-conjugate (normal, if also a subgroup) iff alpha trivial. In
coding the types of decomposition, I assign a weight of "4" to
non-trivial beta, "2" to non-trivial alpha, "1" to non-trivial gamma:
Type 0 (alpha, beta, gamma all trivial): "direct product" +. H and T
are both normal subgroups, operating independently.
Type 1 (only gamma non-trivial): "ordinary product" *, taking a
subscript to mark the particular choice of gamma map if more than one is
possible. The pi product makes T into the quotient group; the order of
its elements under pi has to be multiplied by the order of their internal
carries to find the order in G. For example, Z_4 = Z_2 * Z_2, meaning
that 01 01 = 10, not 01 01 = 00 as in Z_2 + Z_2. This is a seriously
understudied case, given how common it is, driving the proliferation of
subtly distinguishable p-groups. The quaternions Q_8 = Z_2 * (Z_2 + Z_2)
is an instructive example. Write the elements of the first Z_2 (left of
the asterisk) as "+" and "-", of the second as blank and "i", of the
third as blank and "j"; the carry-map is:
i i = - (internal to i)
j j = - (internal to j)
j i = - i j (*induced* carry)
This is what the "undecorated" Z_2 * (Z_2 + Z_2) ought to be reserved
for. Other non-trivial carry maps are possible: but if there is no
induced carry, the group is the rectangular abelian, better written as
Z_4 + Z_2 = (Z_2 * Z_2) + Z_2; while if j induces a carry when passing i but
does not carry internally, then j acts by inversion on the cyclic-4
generated by i and this is D_8 better written Z_4 X| Z_2. The
quaternions are *completely ordinary*, with the carry-map non-trivial in
every case where it can be-consistent with the requirements of
associativity, which force some carries trivial. For example, in
Z_2 * (Z_4 + Z_2) = Z_2 * ((Z_2 * Z_2) + Z_2), calling the generator of
the cyclic-4 in the right factor "i" (it is order-8 in the full G), then
if j induces carry
j i = - i j
we must have
j i^2 = (- i j) i = + i^2 j
so j cannot induce carry on the high-order bit.
Type 2 (only alpha non-trivial): "semidirect product" X| with
superscript for choice of alpha if necessary. H normal, T a subgroup
acting homomorphically on H. In many cases, isomorphic subgroups of
Aut(H) will act to give non-isomorphic semidirect products. To decide
which of these the "undecorated" semidirect product "ought" to stand for,
is a difficult part of this project. For example, there are Z_8 X|^7 Z_2,
Z_8 X|^5 Z_2, and Z_8 X|^3 Z_2 using actions that send the generator of the
cyclic-8 to its 7th, 5th, or 3rd power, with distinct groups resulting.
The action ^7 is most "kindly": a "kind" of group is a parameterized
series, with relations definable by some template, as g^n = id defines
the cyclic kind Z_n, and r^n = t^2 = (tr)^2 = id defines the dihedral
kind D_2n; a decomposition D_2n = Z_n X| Z_2 exists for all in this kind.
So the others should be named some different way:
Z_8 X|^5 Z_2 = Z_2 * (Z_4 + Z_2) and
Z_8 X|^3 Z_2 = Q_8 X| Z_2.
This is typical if H has a center: unless the center is elementary, some
central subgroup must be fixed by the action, and can be pulled out as an
ordinary-product left-factor; if this is not completely ordinary, there
will be some rearrangement. If there is no center, H typically has a
lot of inner automorphisms, and if T contains any inner actions these can
be removed as a direct-product factor; Out(H) is generally quite small.
If the action is free on an elementary center, it is tightly constrained,
but there is often an ambiguity between a *wreathlike* and non-wreath
actions: when H is further decomposed into H_L and H_R (left, right
factors), the action of T on H_R may involve a non-trivial H_L component,
a *carry-back*, and more troublesomely for calculations, the action of T
on H_L may involve a non-trivial *carry-up* to H_R, which defines a
wreathlike action. I use S for "swap", or integral-sign when available,
to distinguish
(Z_3 + Z_3) S Z_2, acting by 01 <-> 10, from
(Z_3 + Z_3) X| Z_2, acting by 11 <-> 22.
Type 3 (alpha and gamma non-trivial): "indirect product" *|, perhaps
needing both subscript and superscript. Usually this can be *rectified*
with a better choice of T to trivialize one or both maps. This is the
smallest example of an unrectifiably indirect decomposition: A_6 *| Z_2
indicates an order-FOUR (at least) automorphism (the order in the right
factor must be multiplied by the order of its carry) whose square is
inner to A_6; now this could be:
"even inner", like conjugation by (1 2 3 4)(5 6); but this is
isomorphic to A_6 + Z_2 mapping (sigma, z) where z is the non-trivial
element of T to (sigma times (1 2 3 4)(5 6), z') where z' now acts
trivially and does not carry.
"odd inner" (inner to S_6, though outer to A_6), like conjugation by
(1 2 3 4); but this can be rectified in the same way, now z' acts by
conjugation by (5 6) which is the "kindly" action like all other cases
of A_n X| Z_2 = S_n.
"even outer", an outer automorphism of S_6 which respects the two
5-cycle conjugacy subclasses in A_6; but some of these are involutions,
so this rectifies to an A_6 X| Z_2 with unkindly (very unique) action.
"odd outer", from Out(S_6) but swapping the 5-cycle conjugacy
subclasses; none of these are involutions, so this is an indirect product
and cannot be made "prettier".
An infinite-group example: F(Q) is the group of "field operations"
(additions, multiplications and their compositions) over the rationals,
decomposing as
F(Q) = Add(Q) X| Mult(Q)
but Mult(Q) has a subgroup Mult^2(Q) restricted to multiplications by
perfect squares, yielding F^2(Q) = Add(Q) X| Mult^2(Q). F^2(Q) is normal
in F(Q) so:
F(Q) = F^2(Q) *| E_2^infinity
where the right factor is an elementary-2 group of infinite dimension
(one for each prime), acting non-trivially on the additives and carrying
into the multiplicatives.
Type 4 (only beta non-trivial): "reverse semidirect product" |X. This
time T is normal, and H |X T = T X| H. The notational distinction can be
used to write Z_2 |X A_6 for the "unkindly" action of an even-outer
automorphism; this is a waste-basket case I use for ambiguous semidirects
I have no better way to mark.
Type 5 (beta and gamma non-trivial): "abnormal product" ~. In every
case without exception, HT has a normal subgroup but neither H (not
normal) nor T (self-conjugate but not a subgroup) is it: if Image(gamma)
non-trivially intersects any conjugate, it shares that intersection with
all conjugates and this is normal; if its conjugates are a
trivial-intersection set, the complement (plus the identity) is a normal
subgroup. Thus this product should never be used at all.
But in defining a "canonical" decomposition for each group, the possible
series of refinements to atomic are ranked in a *lexical* order by the
Type codes at each step, so that for example pulling out a Type 0
direct-product factor as the first step is always favored when possible:
Z_6 could be written Z_2 * Z_3 but Z_2 + Z_3 is objectively better (if
the group-elements are being stored in a computer, "better" means "fewer
calculational steps"). Now, this lexical ordering is modified by
reassigning some decompositions whose maps indicate a different Type to
Type 5 "abnormal": an ordinary product which is incompletely ordinary has
*abnormal carry*; a semidirect or indirect product which my (yet to be
rigorously defined) rules single out as "unkindly" has *abnormal action*;
in either of these cases, Type 5 has the effect of disfavoring this route
of decomposition over any other that uses a normal subgroup as a factor
(but for exploring subgroup structure, knowing of the existence of
non-canonical decompositions like D_8 = Z_2 *~ (Z_2 + Z_2) is helpful).
However, if the group is simple so that only Type 6 or 7 decompositions
are possible, assigning Type 5 *standard abnormal* (naturally, every
decomposition of a simple group is "abnormal") has the effect of singling
it out as favored.
Type 6 (alpha and beta non-trivial): "set product", no symbol but
concatenation. Both H and T are subgroups, but neither is normal, and
neither of the actions is homomorphic. For example, A_5 = A_4 Z_5:
multiply any sigma in A_5 by whatever power of the 5-cycle returns number
5 back home; then sigma is the product of the residual A_4 element and
the inverse power of the 5-cycle; this works for any odd-numbered
alternating group. Among the evens, but A_6 = D_10 F^2(9) representing
F^2(9) (additives by square-multiplicatives in GF(9), order-36) by
<(1 2 3), (4 5 6), (1 4)(2 5 3 6)>.
Few simple groups have a set-product decomposition: an important result
is that PSL(2,q) has a set product iff q = 4, 5, 7, 9, or 11.
Type 7 (all non-trivial): "pseudo-product", sometimes I mark as psi, but
usually I only deal with the standard abnormal marked ~. Here T is not
even a factor-group, but only a *pseudo-group* (not like a semigroup,
which has associativity but not invertibility; a pseudo-group has
invertibility but not associativity). For PSL(2,F) over any (finite or
infinite) field, acting on the projective field F U {infinity}, define
Add(a) : x -> x+a, fixing infinity (a in F; Add(0) = id; Add(infinity)
not defined)
Mult(m) : x -> xm, fixing infinity and 0 (m square; Mult(1) = id;
Mult(0), Mult(infinity) not defined)
Twist(t) : x <-> t + (t-x)^-1, x <-> infinity (t in F; Twist(infinity)
another synonym for id)
PSL(2,F) = (Add(F) X| Mult^2(F)) ~ Twist(F) writing every element as a
product Add(a) Mult(m) Twist(t), for finite fields this gives q times
either q-1 or (q-1)/2 times q+1 choices.
In the representation on the 2-dimensional vectors,
Add(a) is the matrix [1 a] [0 1],
Mult(m) is [m^(-1/2) 0] [0 m^(1/2)],
Twist(0) is [0 -1] [1 0] with all other (non-trivial) Twist(t) obtainable
as the conjugates of Twist(0) by Add(t).
The group product rules are all obtainable from:
Add(a) Add(b) = Add(a+b)
Mult(m) Mult(n) = Mult(mn)
Add(a) Mult(m) = Mult(m) Add(am)
Twist(t)^2 = id (all are their own inverses)
Twist(t) Add(a) = Add(a) Twist(t+a) (alpha trivial on the additives)
Twist(0) Mult(m) = Mult(m^-1) Twist(0) (alpha by inversion on the
multiplicatives; but there is induced carry from Twist(t) if t nonzero)
[Add(1) Twist(0)]^3 = id (a somewhat difficult induced-carry map results)
This presentation helps to clarify the conjugacy classes (everything can
be conjugated to the form Add(chi) Twist(0) where chi is the character)
and other calculational exercises.
Similarly in PSL(3,q) the pseudo-group contains 1 (the identity) plus q
(Twists all conjugates of each other by an additive subgroup) plus q^2
(Twists conjugated by the product of additive subgroups in the same
column per the matrix representation). This kind of rectified
pseudo-product (rectified in the sense that many T-elements are made
conjugates by some subgroup so that alpha is trivialized "by decree"
to that extent) is defined as *diagrammatic* (for all Chevalley groups,
whether classical, exceptional, or twisted, the pseudo-group factors have
a roughly similar structure). It can be proven without any assumptions
about the internal structure of a finite simple group that it has a
standard-abnormal rectified pseudo-product sharing at least some of the
diagrammatic properties; those that are not quite diagrammatic are lumped
as *alternative* (the alternating and sporadic groups have these). The
standard-abnormal decomposition defines a relationship to a smaller
simple group, the *simple core* (for PSL(2,q) this would only be a
prime-cyclic; for larger diagrammatics, a group whose diagram has one
fewer node). The larger simple group is then viewed as a *diagrammatic
extension* or *alternative extension* of the simple core.
The "grail" of this project then is a more streamlined proof to the
Classification Theorem for Finite Simple Groups, with a big lemma to the
effect "a simple core can only support an alternative extension if it
itself has a decomposition of set-product or alternative type". The
alternating groups have alternative extensions, each to the next-higher
A_n+1; only a couple other small cases, the order-168 and order-660
groups, have escapes out of the diagrammatic superfamily, into the Mathieu
and Janko groups, which continue to spawn alternative extensions up to
where all paths merge into the Baby Monster, which has one more
alternative extension to the Monster, which has no finite alternative
extension (I believe there is a horrid infinite simple group, the "Affine
Monster", which I see through a glass darkly).
More information about the Forum
mailing list