> < ^ Date: Thu, 09 Jul 1998 14:06:21 +0200 (CEST)
> < ^ From: Thomas Breuer <Thomas.Breuer@Math.RWTH-Aachen.DE >
< ^ Subject: Re: generators of SL(2, GF(q)) ?

Dear GAP Forum,

some remarks may be of interest in addition to Alexander's answer
to Guido Helmers' question

My problem is the following:

I try to find all the triples of generators of the two-dimensional
special linear group over a finite field (say G:= SL(2, GF(p^n))),
upto automorpism of these triples, and such that the product of the
three generators equals 1; my question is:

For a fixed finite group G (in my case a matrix group); a fixed number m
(in my case 2 or 3); and fixed numbers a1,..am (dividing Size(G)),
what is the quickest way to find, for example, the set
(1) {{g1,..,gm} \in Gx..xG| <g1,..,gm>=G and ord(gi)=ai for all i}
(2) {{g1,..,gm} \in Gx..xG| 
              <g1,..,gm>=G and g1...gm=1 and ord(gi)=ai for all i} 

or, if no function exists which returns such m-tuples given a1,..,am
(3) to find the set of ALL m-tuples of generators of G

(a) There are character theoretic methods to compute the number of
m-tuples (g1,g2,...,gm) with product g1 g2 ... gm the identity
and each gi chosen from a given subset of G (for example,
a certain conjugacy class or the set of all elements of a given
order ai in G), see [1] for this and related problems.

(b) Clearly (a) does *not* solve the problem to find the *generating*
m-tuples with the prescribed properties, or their number
(which might be more reasonable for larger groups).
But since Aut(G) acts semiregularly on the sets defined in (1) and
(2) above, one can conclude that the set in question is empty if
the number in (a) is less than the order of Aut(G).
A less obvious and very useful such criterion of nongeneration is
described in [2].

(c) If one has more information about the group G and its subgroups,
such as the primitive permutation characters of G or the character
tables of all (conjugacy classes of) maximal subgroups of G,
then it might be possible that character theoretic methods suffice
to prove that the set (1) or (2) is nonempty.
Again, this does in general *not* solve the problem to compute
the cardinality of the set in question (or the set itself).

(d) For the groups SL(2,q), q a prime power, it may be possible to
treat (a)-(c) using the generic character table(s).
This way one might obtain results for all values of q, not only
for one specific group.

[1] G. A. Jones, Enumeration of Homomorphisms and Surface-Coverings,
    Quart. J. Math. Oxford Ser. (2), no. 46 (1995), 485--507.

[2] L. L. Scott, Matrices and cohomology,
    Annals of Mathematics, no. 105 (1977), 473--492.

The methods sketched above can be combined with the solution given
by Alexander in the sense that they help to avoid unnecessary loops
over m-tuples of group elements.
But since my remarks do not address exactly the question stated by
Guido, I do not want to tell the details here.

I have GAP programs for dealing with the questions (a)-(c),
if someone is interested in this then (s)he can ask me directly.

Kind regards
Thomas Breuer

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