Bounds on the number of lifts of a Brauer character in a p-solvable group
Abstract The Fong–Swan theorem shows that for a p-solvable group G and Brauer character φ ∈ IBr p ( G ) , there is an ordinary character χ ∈ Irr ( G ) such that χ 0 = φ , where 0 denotes restriction to the p-regular elements of G. This still holds in the generality of π-separable groups, where IBr p ( G ) is replaced by I π ( G ) . For φ ∈ I π ( G ) , let L φ = { χ ∈ Irr ( G ) | χ 0 = φ } . In this paper we give a lower bound for the size of L φ in terms of the structure of the normal nucleus of φ and, if G is assumed to be odd and π = { p ′ } , we give an upper bound for L φ in terms of the vertex subgroup for φ.
Partial characters of π-separable groups
Our concern in this expository paper is the character theory of a finite group G as seen from the perspective of a set π of prime numbers. There are no new results here and few new ideas. Our purpose is to present in as accessible a manner as possible, the proofs of some theorems in the character theory of π-separable groups, and to explain the significance of these results.
Character Theory of Finite Groups
Algebras, modules, and representations Group representations and characters Characters and integrality Products of characters Induced characters Normal subgroups T.I. sets and exceptional characters Brauer's theorem Changing the field The Schur index Projective representations Character degrees Character correspondence Linear groups Changing the characteristic Some character tables Bibliographic notes References Index.