Aschbacher Analysis

The basic facilities provided by Magma for computing with matrix groups over finite fields depend upon being able to construct a chain of stabilizers. However, there are many examples of groups of moderately small degree where we cannot find a suitable chain. An on-going international research project seeks to develop algorithms to explore the structure of such groups. The main theoretical underpinning of the project comes from the classification by Aschbacher (1984) of the (maximal) subgroups of $\mbox{\rm GL}(d,q)$ into nine families. Much of the research effort to date has been devoted to designing algorithms to decide whether G belongs to one of the eight families whose members have a normal subgroup preserving a ``natural linear structure"; here, we plan to exploit this information to explore G further, ultimately producing a composition series for G.

• Determine whether a group preserves a form modulo scalars.
• The Niemeyer-Prager classical group recognition algorithm as implemented in Magma by Alice Niemeyer and Anthony Pye.
• Determine whether a subgroup G of $\mbox{\rm GL}(d,q)$ acts imprimitively on the underlying vector space. a block system, respectively.
• Test whether a matrix group G acts as a semilinear group of automorphisms on some vector space.
• Test whether a matrix group G preserves a non-trivial tensor product decomposition.
• Test whether a matrix group G is tensor-induced.
• Search for decompositions (corresponding to certain Aschbacher families) with respect to the normal closure of a supplied subgroup.
• The Glasby-Howlett algorithm to decide if the absolutely irreducible group $G \leq \mbox{\rm GL}(d, K)$ has an equivalent representation over a subfield of K.
• Given a group G of $d \times d$ matrices over a finite field E having degree e and a subfield F of E having degree f, write G as a group generated by the matrices of G written as $d e/f \times d e/f$ matrices over F.