Study Onπ-block Theory

Based on the existing results in the literature, further research onπ-block theory is carried on in this paper and related results on several aspects are obtained which we state as follows:In chapter 2, two correspondences in the set of theirπ-special characters andπ-Brauer characters are presented in two homogenous groups. In chapter 3, the transitive property of coverage is proved and it is deduced that B lies over a single orbit of Irr(N) and there is exactly oneπ-block of G which coversφ, if N(?)G is aπ'-subgroup where B∈Blkπ(G),φ∈Irr(N), G is aπ-separable group. In character 4, based on the definition of defect groups ofπ-separable groups, some related results are presented about defect groups and it is shown thatδ(B)∈Hallπ(G) if and only if there existsχ∈Irr(B) such that (χ(1),p)=1,for any p∈π.In particular,δ(B0) is a Hallπ-subgroup of G, where Bo is the mainπ-block of G.Further more, some related results are deduced based on Brauer's first main lemma.