# Research problems

Revision of August 2015.

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 \(L_2(17)\),
\(L_2(41)\), Sz(8), \(U_3(8)\), and \(U_3(4)\),
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.
The case of \(L_2(41)\) is particularly interesting, as we thought it had been
eliminated by theoretical work, but a subtle error came to light
after many years,
re-opening the possibility of a maximal subgroup \(L_2(41)\).
I have now done this case, and found a maximal \(L_2(41)\).
Update August 2015: I have eliminated the case Sz(8).
On the other hand, the cases \(L_2(8), L_2(13), L_2(16)\) have not been
published, and should probably be done again.

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 have done this (up to dimension 12)
and written a book on
the results and methods. Colva's student Anna Schroeder extended it up to dimension 15,
and work is under way to extend this further.
One should probably aim for dimensions up to 250 in the not-too-distant future.

Current list of incomplete cases is: HN, \(Fi_{23}\), \(
Fi_{24}'\), \(Co_1\), and B (and of course M).
A fairly straightforward computational project, except for the Monster case
which will be hard.

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 idea of using involution centralizers in black-box groups
(especially groups of characteristic p, p odd) was developed by
Altseimer and Borovik to distinguish \(\mathrm{P}\Omega_{2n+1}(q)\) from \(\mathrm{PSp}_{2n}(q)\)
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.)

A nice construction `by hand', for example in 56 dimensions in characteristic 0,
would be good to have.

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's PhD thesis (available from my research webpage) considers 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 \(G = D_{4pq}, A = D_{2p}\)
and \(B = D_{2q}\) 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 \(E_7(2)\), and dealt with most of
the cases: a few small non-local cases remain.
A combination of theoretical and computational work will be required.
The case \(E_7(2)\) has been completed by Ballantyne, Bates and Rowley.

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.

This is the last remaining case of multiplicity-free characters
in which the centralizer algebra is not known. The quotient action
of B on \(Fi_{23}\) 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 \(2 \times 10^{15}\) points).

The Lie algebras of types \(G_2, F_4\) and \(E_8\) have respective dimensions
14, 52 and 248. Observe that \(14=2(2^3-1)=2(1+2+2^2),
52=2(3^3-1)=4(1+3+3^2)\) and \(248=2(5^3-1)=8(1+5+5^2)\).
This is related to the existence of subgroups \(2^3\cdot L_3(2)\) in
\(G_2(\mathbb C)\), \(
3^3{:}L_3(3)\) in \(F_4(\mathbb C)\), and
\(5^3{:}L_3(5)\) in \(E_8(\mathbb C)\).
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 \(G_2\) in a paper of mine published in Math. Proc. Cambridge Philos. Soc.,
\(F_4\)
and \(E_6\) (preprint); work in progress by Nicholas Krempel for \(E_8\).)
A similar construction of \(E_8\) using the subgroup \(2^{5+10}L_5(2)\)
is given by Johanna Rämö in her PhD thesis.

The Freudenthal magic square gives some tantalizing hints
about the real nature of the groups/Lie algebras of exceptional type. The exceptional
Jordan algebra shows how express \(F_4\) and \(E_6\) in terms of \(3\times 3\)
matrices over octonions. What is the right way to extend this to \(E_7\) and \(E_8\)?
Is there some meaning to the following pattern: \(F_4\) has dimension 26 over the real
numbers, \(E_6\) has dimension 27 over the complex numbers, \(E_7\) has dimension 28
over the quaternions, and \(E_8\) has dimension 31 over the octonions?
(I have obtained a quaternionic construction of \(E_7\) - see my webpage for a preprint.)

The matrix group recognition project led by Leedham-Green has been very successful at
developing algorithms to recognise arbitrary groups given as sets of generating matrices.
There are still a few gaps in the algorithm, however, one of the most serious of which
is that of "strong" recognition of finite simple groups of exceptional type. There
are various algorithms required, all of which require intimate knowledge of the groups
themselves and their subgroup structure. The right way forward seems to be to
reduce to smaller (usually classical) cases by using involution centralizers
(in odd characteristic) or analagous methods in characteristic 2. This leaves a
few base cases where these methods do not apply, and some more serious work is
required. See Henrik Bäärnhielm's thesis and publications for some
examples.

# Archive of solved problems

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 left quite a lot of work to
do here, especially for quasisimple groups (i.e. proper covering groups of simple groups).
This project has been completed by Allan Steel, who has written programs in MAGMA
which permit more-or-less automatic computation of everything on this list.

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. This has also been completed by Allan Steel.

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 ^{3}D_{4}(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 ^{3}D_{4}(q)
for which generic character tables are known. Thus it is in principle a
straightforward calculation to complete this case.
This problem has been solved by Vdovin and Gal't.

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!).
This will then provide an existence proof of M, independent
of that of Griess. Adeel Farooq has done this in his PhD thesis.

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(2^{2}) and N(2^{5}) in the Baby Monster, where the minimal base size
is either 2 or 3 (probably 3). Both of these have now been resolved, in work with
Eamonn O'Brien, Max Neunhöffer and Felix Noeske.

* This page is maintained by ***Robert Wilson**. The
views and opinions expressed in these pages are mine. The contents of these
pages have not been reviewed or approved by Queen Mary, University of
London.