Computational Algebra Seminar: Glasby -- The probability that two elements with large 1-eigenspaces generate a classical group

Speaker: Stephen Glasby  
Title: The probability that two elements with large 1-eigenspaces generate a classical  
 group  
Time & Place: 15.00-16.00, Thursday 21 May, SMRI Seminar Room  
Abstract: There has been much research, by many different authors, on
matrix group algorithms.  Given a finitely generated subgroup G of a finite general
linear group GL(d,q), the first step is to find a composition series for G, this
involves finding the simple groups from which G is "built".  After chopping G into
indivisible parts, one has to "recognize" the simple groups.  The most common
(non-cyclic) simple groups to recognize are the classical groups, and the core case
involves recognizing classical groups acting on their natural module.  Existing
algorithms construct a naturally embedded classical subgroup H of G which is generated
by new generators g1, g2 which have large 1-eigenspaces.  We prove that we can find g1,
g2 withhigh probability, and that g1, g2 generate H with probability at least 0.97.  

This result appears in an 80 page preprint with Alice Niemeyer (Aachen) and Cheryl
Praeger (UWA), and took us about 6 years to complete.  I will give an overview of the
working parts of this project.