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.
Inductive pairs and lifts in solvable groups
Abstract The Fong–Swan theorem showed that for each irreducible Brauer character ϕ of a finite solvable group, there exists at least one ordinary irreducible character χ that lifts ϕ. Recently there has been some progress toward understanding the set of lifts of a given Brauer character, by constructing certain canonical sets of lifts and studying their properties. In this paper we develop the theory of lifts of Brauer characters that are induced from inductive pairs. This leads to a lower bound on the number of ‘well-behaved’ lifts of a given Brauer character in terms of the inductive pair. Moreover, it is shown that given a ‘well-behaved’ lift, the vertices for the lift (a generalization of the vertices of the corresponding Brauer character) are unique up to conjugacy.
On Hilbert's thirteenth problem and related questions
Hilbert's thirteenth problem involves the study of solutions of algebraic equations. The object is to obtain a complexity estimate for an algebraic function. As of now, the problem remains open. There are only a few partial algebraic results in this connection, but at the same time the problem has stimulated a series of studies in the theory of functions with their subsequent applications. The most brilliant result in this cycle is Kolmogorov's theorem on superpositions of continuous functions.