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.
Navarro vertices and normal subgroups in groups of odd order
Let p be a prime and suppose G is a finite solvable group and χ is an ordinary irreducible character of G. Navarro has shown that one can associate to χ a pair (Q, δ), where Q is a p-subgroup of G and δ is an irreducible character of Q, which is unique up to conjugacy. This pair is called the Navarro vertex of χ, and in this paper we study the behavior of the Navarro vertices of χ with respect to normal subgroups of G when G is assumed to have odd order and χ is a lift of an irreducible Brauer character of G. For NCG, we use these results to give a sufficient condition for the constituents of χN to be lifts of Brauer characters, and we develop a Clifford-type correspondence for lifts with a given Navarro
Decidability of some intuitionistic predicate theories
Suppose T is a first order intuitionistic theory (more precisely, a theory of Heyting's predicate calculus, e.g., abelian groups, one unary function, dense linear order, etc.) presented to us by a set of axioms (denoted also by) T. Question. Is T decidable? One knows that if the classical counterpart of T (i.e., take the same axioms but with the classical predicate calculus as the underlying logic) is not decidable, then T cannot be decidable. The problem remains for theories whose classical counterpart is decidable. In [8], sufficient conditions for undecidability were given, and several intuitionistic theories such as abelian groups and unary functions (both with decidable equality) were shown to be undecidable. In this note we show decidability results (see Theorems 1 and 2 below), and compare these results with the undecidability results previously obtained. The method we use is the reduction-method, described fully in [12] and widely applied in [3], which is applied here roughly as follows: Let T be a given theory of Heyting's predicate calculus. We know that Heyting's predicate calculus is complete for the Kripke-model type of semantics. We choose a class M of Kripke models for which T is complete, i.e., all axioms of T are valid in any model of the class and whenever φ is not a theorem of T, φ is false in some model of M.