> < ^ From:

< ^ Subject:

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

http://www.math.colostate.edu/~hulpke

> < [top]