Usage of computer algebra packages for solving difficult mathematical problems

The talk propagates using the computer algebra packages (systems for symbolic coputations), which have got world-wide extension for the last years, for solving difficult mathematical problems of discrete nature. The speaker intends to describe his experience in usage of the GAP package (Groups, Algorithms and Programming) for the complete solution of several long-standing problems concerned with the arrangement of subgroups in finite groups. An optimal join of theoretical and computer-algebraic methods has allowed the author to cover several conjectures mentioned in the following well-known monographies:
1. B. Huppert. Endliche Gruppen I. Springer-Verlag, 1967;
2. K. Doerk, T. Hawkes. Finite Soluble Groups. Walter de Gruyter, 1991,
as well as a series of problems on polynormal subgroups posed by Z. I. Borevich nearly 20 years ago. An efficient computational tool for investigation of such problems was found out, that is the table of marks (or Burnside matrix) method.
This talk reflects the experience of working with such packages, which was accumulated for three years period of existence of St. Petersburg seminar on Computer Algebra at Steklov Math. Institute (have a look at http://gauss.pdmi.ras.ru/~vimys/seminar/). The computer algebra packages available in PDMI (GAP, MuPAD, Mathematica, Maple, Axiom, Pari, etc.) are discussed and certain their features are mentioned.