Main field of research: group theory and its applications, computational group theory, linear groups, subgroups of algebraic groups.
1.1 Computing in nilpotent
matrix groups.
A new approach to computing in nilpotent matrix groups has been developed.
The following algorithms have been obtained: (i) algorithms testing nilpotency
of matrix groups (ii) algorithms testing irreducibility/primitivity of a
nilpotent matrix group G over a finite field, and algorithms constructing
irreducible G-modules and nonrefinable G-systems of imprimitivity. Additionally,
algorithms return detailed structural information about nilpotent linear
groups.
1.2 Deciding
finiteness for matrix groups.
Deterministic polynomial time algorithms for deciding finiteness of matrix groups over a field of transcendence degree one over a finite field have been designed.
1.3 Computer
databases for matrix solvable groups.
Algorithms constructing irreducible maximal solvable subgroups of the special linear group SL(q, k) have been developed (here q is a prime number, k is a finite field). The algorithms have been implemented in the group theory computer system GAP as a group library. Functions of the library return SL(q, k)-conjugacy class representatives of irreducible maximal solvable subgroups of SL(q, k). For each subgroup the algorithms construct matrices of a (consistent) polycyclic presentation. The database is available here .
2 Linear groups. Subgroups of linear algebraic groups.
2.1 Nilpotent primitive irreducible linear groups over a finite field have been classified up to conjugacy; algorithms providing electronic access to the classification have been developed.
2.2 Irreducible
maximal solvable subgroups of prime degree classical groups over an arbitrary
field have been classified up to conjugacy (about 120 pp. in Russian; in English
Siberian Math. J.
35 (1994), no. 2, 286—293 and Siberian Math. J.
33 (1992), no. 6, 973--979 (1993); see also a preprint here
or here:
2.3 The classification up to conjugacy of absolutely irreducible maximal periodic subgroups of isometry groups U(f, k), SU(f, k) has been obtained (here f is a nondegenerate Hermitian or bilinear symmetric form over a field k of odd characteristic) (published in Russian).
2.4 Maximal periodic subgroups of the projective linear group PGL(q, F) over a field of positive characteristic has been classified up to conjugacy (q is a prime number).
2.5 The classification of maximal solvable subgroups of Chevalley groups of type G2 field has been obtained (preprint).
2.6 Some classes of maximal periodic subgroups of the isometry group U(f, k) have been described (f is a nondegenerate bilinear form over a field k of characteristic 0) (published in Russian).
3 Current research
(1) CGT. Algorithms for computing in matrix groups over integral domains; algorithms constructing irreducible solvable matrix groups over finite fields.
(2) Linear groups. Linear nilpotent groups: structural, classification, and asymptotic results (bounds on number of generators, number of conjugacy classes etc.). This research is motivated by its application to (1).
(3) Algebraic groups. Solvable and nilpotent subgroups of linear algebraic groups. This research is partially based on (2).
My collaborator: Dane Flannery