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.
Lifts of Partial Characters with Respect to a Chain of Normal Subgroups
In this paper, we consider lifts of π-partial characters with the property that the irreducible constituents of their restrictions to certain normal subgroups are also lifts. We will show that such a lift must be induced from what we call an inductive pair, and every character induced from an inductive pair is such a lift. With this condition, we will get a lower bound on the number of such lifts.
Normed units in abelian group rings
We compute explicitly the isomorphic structure of the normalized unit group of an abelian group ring under some minimal natural restrictions on the group basis and the coefficient ring. This enlarges affirmations due to Chatzidakis-Pappas (J. London Math. Soc., 1991) and Mollov (Publ. Math. Debrecen, 1971).