Research problems
So far there are 43 known conjugacy classes of maximal subgroups of the
Monster sporadic simple group. To complete the classification of
maximal subgroups, it is necessary and sufficient to classify
the subgroups isomorphic to Sz(8), U3(8), and U3(4),
and their normalizers. In principle, Beth Holmes and I are working
on this, but progress is slow. A combination of theoretical and
computational work is required.
Richard Barraclough's PhD thesis includes a classification of
all nets which are centralized by an element of prime order p, for
p at least 5. We wish to extend this to a complete list of all
the nets (there are more than 13575 of them altogether).
The nets are geometrical objects defined by triples of 2A-elements
(6-transpositions) in the Monster. If (a,b,c) is such a triple
you can braid the first two elements repeatedly to get (b,a^b,c),
(a^b,b^{ab},c) etc, and you get back to where you started after
o(ab) steps. This corresponds to a polygon with o(ab) sides.
By similarly braiding the other two pairs, and factoring by a
suitable equivalence relation, we get three polygons around a vertex.
The 6-transposition property means that the polygons have at most
6 sides, so either they are all hexagons, and they form
a surface of genus 1, or some are not hexagons, and they form a
surface of genus 0.
Re-do Kleidman's work on maximal subgroups of classical groups
in dimension up to 12 (and possibly more). Needs great care and accuracy.
John Bray, Derek Holt and Colva Roney-Dougal are working on this.
One should probably aim for dimensions up to 250 in the not-too-distant future.
Current list of incomplete cases is: HN, Fi23,
Fi24', Co1, and B (and of course M).
A fairly straightforward computational project.
and perhaps also M. All that remains is the prime 2.
What is required is a detailed analysis of the 2-local subgroups, and
calculation of (at least part of) the character tables of the
stabilizers of the so-called radical p-chains. The problem is that these
subgroups are large and complicated, and many of them have not
been constructed at all, or do not have representations
that are small enough to study easily.
The list of embeddings is in a paper by Hiss and Malle in the LMS JCM,
but there are a few errors in this list. Simon Nickerson computed a large number of
these embeddings in his PhD thesis (2005), but there is still a lot of work to
do here, especially for quasisimple groups (i.e. proper covering groups of simple groups).
The idea of using involution centralizers in black-box groups
(especially groups of characteristic p, p odd) was developed by
Altseimer and Borovik to distinguish O2n+1(q) from PSp2n(q)
in Monte Carlo polynomial time. Parker and Wilson used the same basic
ideas to distinguish between a simple group and a non-simple group.
However, they did not develop a proper algorithm, let alone
implement it: these need to be done. Also, a complexity analysis
of such an algorithm will require a careful study of properties of
involutions and their centralizers in groups of Lie type. (An algorithm,
with complexity analysis, has now been developed, but not implemented.)
This has been complete for the simple groups, but not for all the
automorphism groups or (especially) the covers. There are some hard cases here,
for example 3.ON, where I do not have a project plan for constructing the
representations.
A nice construction `by hand', for example in 56 dimensions in characteristic 0,
would be good to have.
This is Problem 14.82 from the Kourovka Notebook.
A recent paper by Tiep and Zalesskii classifies the real simple groups
(i.e. the simple groups all of whose elements are conjugate to their inverses).
This reduces the problem to looking at certain orthogonal groups and
the triality twisted groups 3D4(q).
Using the well-known classification of semi-simple classes and their
centralizers, it should be possible to determine case-by-case which
of the real elements are strongly real (i.e. are the product of
two involutions). Orthogonal groups in odd characteristic were apparently
already done. Orthogonal groups in characteristic 2 have been done by
Johanna Rämö. This leaves just the triality groups 3D4(q)
for which generic character tables are known. Thus it is in principle a
straightforward calculation to complete this case.
Both questions involve looking at groups of the form V.G where
G is simple and V is an irreducible G-module, and seeing if there
are elements of larger order in V.G than there are in G.
Most of the time there probably are, but potentially this
might require a huge knowledge of modular representation theory of
all the finite simple groups in order to sort it all out.
Kourovka Notebook 14.69. Various questions about how many involutions
you need, subject to various conditions, in order to generate a
finite simple group. Some of these questions have been answered by
Nuzhin and others, but some of the trickier ones remain open.
Jonathan Ward is working on the case where all the involutions are conjugate,
and their product is 1. Here some groups need 6 involutions, and others
can be generated by 5.
If G = AB where A, B are finite groups of co-prime orders, is it true
that k(G) is at most k(A).k(B), where k(G) denotes the number of
conjugacy classes in G? If either A or B is normal in G, this is
straightforward to prove. It is not true if the co-prime
condition is dropped: take G = D4pq, A = D2p
and B = D2q where p, q are distinct odd primes.
The general problem is appealingly easy to state, but looks
horribly hard to attack.
Chris Bates (Manchester) has looked at E7(2), and dealt with most of
the cases: a few small non-local cases remain.
A combination of theoretical and computational work will be required.
One reason for wanting these is to assist in the classification of nets
(Problem 2 above), though I think all the ones required for that purpose
are known by now. This problem also relates to Problem 6.
Determine exactly which finite simple groups can be generated by elements
x (of order 2) and y (of order 3) with xy of order 7.
The alternating groups were done by Higman and Conder. The sporadic groups
are done by various authors. Lucchini, Tamburini and JS Wilson have
shown that classical groups in large dimensions are always Hurwitz groups.
Some small rank classical and exceptional groups have also been done.
But there is a huge gap in between where an enormous amount remains to be
done. Sun (student of JS Wilson) filled in a significant part of this,
as did Vsemirnov, but the problem is still wide open.
Working with 196882 x 196882 matrices over GF(3), find elements of the
Monster which satisfy Norton's presentation. Verify these relations
on the whole space (requires a substantial amount of computer time!).
I think this will then provide an existence proof of M, independent
of that of Griess. Adeel Farooq is working on this.
This is the last remaining case of multiplicity-free characters
in which the centralizer algebra is not known. The quotient action
of B on Fi23 has been dealt with, so it only (!) requires
some sign problems to be sorted out. However, there is a difficulty
in finding a way to calculate in this permutation representation
(on about 2 x 1015 points).
John McKay asked me this question. He wants to know the exact numbers of
conjugacy classes in all the finite simple groups. Also the exact numbers of
classes of cyclic subgroups. The latter seems to me to be impossible
to answer, as it depends on factorising all sorts of numbers as products
of primes. The former should be possible to answer in some way, as
there are reasonable parametrisations of the conjugacy classes in all the
finite simple groups.
A base for a permutation group is an ordered set of points whose
joint stabilizer is trivial.
Joint work with Tim Burness and Eamonn O'Brien determines the minimal base size for all
primitive permutation representations of sporadic simple groups, with two exceptions.
These are the groups N(22) and N(25) in the Baby Monster, where the minimal base size
is either 2 or 3 (probably 3). One of these has now been resolved, in work with
Eamonn O'Brien, Max Neunhöffer and Felix Noeske (base size 2), but the other is
still resisting (conjectured base size still 3).
The Lie algebras of types G2, F4 and E8 have respective dimensions
14, 52 and 248. Observe that 14=2(23-1)=2(1+2+22),
52=2(33-1)=4(1+3+32) and 248=2(53-1)=8(1+5+52).
This is related to the existence of subgroups 23L3(2) in G2(C),
33L3(3) in F4(C), and 53L3(5) in E8(C). Find good descriptions of the
Lie algebras from this point of view.
Work of Burichenko, Tiep, and others goes some way to answering this question (see Kostrikin and Tiep,
Orthogonal decompositions and integral lattices). But it would be nice to get constructions which
do not in any way depend on the classical constructions. (Done for G2, F4
and E6; work in progress for E8.)
This page is maintained by Robert Wilson. The
views and opinions expressed in these pages are mine. The contents of these
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London.