> < ^ Date: Mon, 15 Sep 2003 10:44:43 -0600 (MDT)
> < ^ From: Alexander Hulpke <hulpke@math.colostate.edu >
< ^ Subject: Re: Words

Dear Chris Bates,

I have Co_1 represented as a matrix group with dimension 24 over GF(2) (using
the standard generators a & b from the www-ATLAS) and several group elements
in matrix form. I would like to express these elements as words in the
generators. I have tried constucting a free group of rank 2 and creating an
appropriate homomorphism using "GroupHomomorphismByImages" but this exhausts
the workspace I am using. Any ideas about other ways of proceeding?

Essentialy decomposition works via a permutation representation. The default
for this is action on some vectors, so there is no reason for this to be the
representation of smallest degree. This is likely the problem you ran into.

Therefore the first attempt would be to construct (find suitable vectors,
say eigenvectors of random elements) an action homomorphism that will
translate your matrix group to a permutation group of degree ~98000
(smallest deg. perm rep for Co1). With this representation I checked that
GAP (using about 250MB of memory) is able to decompose into generators;
however the resulting words I got were of Length several 100000, which might
not have been, what you had in mind. You can reduce the length a bit by
adding further generators that are known short words, but this will not help
that much, you will have trouble getting below Length a few 1000.

A much better approach for such a group about which you know a lot is to
construct the word ``bespoke'' by hand:
- Determine (trace, Order &c.) the class of this element x.
- Use the ATLAS webpages from Birmingham to find a word for a representative
r of this class
- Now start computing conjugates of r and x by random short words. If you
find a,b s.t. r^a=x^b you found a conjugating element which will give you
a word.
- If you have a small subgroup generated by short words, you can look at
conjugation orbits under this subgroup instead to reduce the search space.

There are various refinements to this technique. The papers of Robert Wilson
on Standard generators (e.g.

Wilson, Robert A.(4-BIRM-SM)
Standard generators for sporadic simple groups.
J. Algebra 184 (1996), no. 2, 505--515.

Suleiman, Ibrahim A. I.(JOR-MUT-MS); Wilson, Robert A.(4-BIRM-SM)
Standard generators for $J\sb 3$. (English. English summary)
Experiment. Math. 4 (1995), no. 1, 11--18.

) describe some such methods.

Best wishes,

Alexander Hulpke

-- Colorado State University, Department of Mathematics,
Weber Building, 1874 Campus Delivery, Fort Collins, CO 80523-1874, USA
email: hulpke@math.colostate.edu, Phone: ++1-970-4914288

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