Csoportok és más algebrai struktúrák = Groups and other algebraic structures

Az absztrakt algebra egyes reszteruletein (a csoportelmeletben, az invariansok elmeleteben, a gyűrűelmeletben, a Lie-algebrak elmeleteben, a hurkok [""loop""-ok] elmeleteben es az univerzalis algebraban) vegeztunk kutatasokat. Eredmenyeinkről 58 dolgozatban es egy konyvben szamoltunk be. Tobb publikacionk vezető matematikai folyoiratokban (Annals of Mathematics, Journal of the European Mathematical Society, Journal of Algebra, Journal of Group Theory, Journal of Algebraic Combinatorics, Proceedings of the American Mathematical Society stb.) jelent meg. Szamos eredmenyunk kozul itt (1) a Lie-tipusu egyszerű csoportok novekedesi fuggvenyeről szolot (Pyber Laszlo es Szabo Endre), (2) a 3x3-as valos szimmetrikus matrixok diszkriminansanak 5 negyzet osszegekent valo felirasat (Domokos Matyas), es (3) a felső haromszogmatrixok csoportjanak konjugaltosztalyszamara vonatkozo Higman-sejtessel kapcsolatosat (Halasi Zoltan es Palfy Peter Pal) emeljuk ki. Az (1) eredmenyt velunk egyidőben Breuillard, Green es Tao is bebizonyitottak. Ennek jelentős kovetkezmenyei vannak az expander grafok teruleten is. A (2) eredmeny Kummer tobb mint masfel evszazados tetelet erősiti, eddig csak 7 negyzet osszegekent valo feliras volt ismert. A (3) eredmeny a Higman-sejtes egy altalanositasat cafolja, ezaltal ketsegesse teve az eredeti sejtes ervenyet is. A kutatasokba harom tehetseges fiatal kutatot is sikerult bekapcsolnunk az OTKA tamogatasaval. | We have conducted research in various subfields of abstract algebra (in group theory, in the theory of invariants, in ring theory, in the theory of Lie algebras, in the theory of loops, and in universal algebra). Members of the research team published 58 papers and a book. Many of our publications appeared in leading mathematical periodicals (Annals of Mathematics, Journal of the European Mathematical Society, Journal of Algebra, Journal of Group Theory, Journal of Algebraic Combinatorics, Proceedings of the American Mathematical Society, etc.). We just mention here our three most important results: (1) on the growth in simple groups of Lie type (L. Pyber and E. Szabo); (2) decomposing the discriminant of a 3x3 real symmetric matrix into the sum of 5 squares (M. Domokos); (3) a result concerning a generalization of Higman's conjecture on the number of conjugacy classes in the group of upper unitriangular matrices (Z. Halasi and P. P. Palfy). Result (1) has been obtained simultaneously by Breuillard, Green, and Tao; it has important implications for expander graphs. Result (2) improves upon a result of Kummer from the middle of nineteenth century; up till now only a decomposition into 7 squares has been known. Result (3) refutes a generalization of Higman's conjecture, hence making the validity of the original conjecture doubtful. The support of OTKA made it possible to employ three talented young researchers as well.