On the Determination of Sylow Numbers

Let G be a finite group and denote by $${\mathbb {Z}}G$$ Z G its integral group ring. In this note we study whether groups with isomorphic integral group ring have the same Sylow numbers. We show that the Sylow q - numbers (i.e. the number of Sylow q - subgroups) coincide provided $${\mathbb {Z}}G \cong {\mathbb {Z}}H$$ Z G ≅ Z H and G is q-constrained. If additionally $$O_{q'}(G)$$ O q ′ ( G ) is soluble the set $$\mathrm{sn}(G)$$ sn ( G ) of all Sylow numbers of G is determined by $${\mathbb {Z}}G.$$ Z G . This holds as well in the cases when G has dihedral Sylow 2-subgroups or when all Sylow subgroups are abelian. G. Navarro raised the question whether even the ordinary character table $$\mathrm{X}(G)$$ X ( G ) of a finite group G determines the Sylow numbers of G. We prove that this is the case when G is nilpotent-by-nilpotent, quasinilpotent, a Frobenius group or a 2-Frobenius group. In particular Sylow numbers of supersoluble groups are given by their ordinary character table. It is proved that for a finite group G with more than two prime graph components $${\mathbb {Z}}G$$ Z G determines $$\mathrm{sn}(G).$$ sn ( G ) . For this we show that almost simple groups of several series of finite groups of Lie type are characterized by their ordinary character table up to isomorphism. An essential tool for many results is that the Sylow numbers of a finite group G are group - theoretically determined from those of factor groups provided G has more than one minimal normal subgroup. In an appendix a short survey on Sylow like theorems for integral group rings is given by the first author. Through all sections several open problems are stated.