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Research interests

  • Computational discrete mathematics and its applications.
  • Abstract algebra, combinatorics, optimisation and search.
  • Open-source mathematical software development.
  • Parallel computing in a variety of settings.
  • Recomputable scientific experiments.

Group Rings

In my PhD thesis, I proved that the normalized unit group of the modular group algebra of a 2-group of maximal class G has a section isomorphic to the wreath product Cwr G’ of the cyclic group of order 2 and the derived group G’ of G, giving for such groups a positive answer on a question formulated by Aner Shalev. Later I gave a construction of the wreath product Cwr G’ for another class of 2-groups.

Later I became also interested in the Modular Isomorphism Problem.

My current interests in the area of group rings are concentrated around torsion units of integral group rings of finite groups. The long-standing conjecture of Hans Zassenhaus (ZC-1) says that every torsion unit in the integral group ring of the finite group G is rationally conjugate to an element in G. Wolfgang Kimmerle proposed to relate (ZC) with some properties of graphs associated with groups. The Gruenberg – Kegel graph (or the prime graph) of G is the graph with vertices labelled by the prime divisors of the order of with an edge from p to q if and only if there is an element of order pq in the group G. Then Kimmerle’s conjecture (KC) asks whether G and the notmalized unit group of its integral group ring have the same prime graph.

Jointly with Victor Bovdi, we started the program of verifying (KC) for sporadic simple groups. Currently we are able to report on the checking (KC) for the following 13 out of 26 sporadic simple groups:

  • Mathieu groups M11M12, M22M23, M24
  • Janko groups J1J2, J3
  • Higman-Sims group HS
  • McLaughlin group McL
  • Held group He
  • Rudvalis group Ru
  • Suzuki group Suz