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.
Induction and restriction of lifts of Brauer characters in groups of odd order
The Fong-Swan theorem shows that if φ is any irreducible Brauer character of a finite solvable group G, then there necessarily exists an ordinary irreducible character χ of G that restricts to φ. In this case we say that χ is a lift of φ. If N G and χ ∈ Irr(G) is a lift, it is also known that the constituents of χN need not be lifts. In this paper we give a sufficient condition for the constituents of χN to be lifts if G is assumed to have odd order. This condition depends on the local structure of G, and in particular the Navarro vertices. We also give sufficient conditions for the constituents of ψG to be lifts if N G and ψ ∈ Irr(N) is a lift. Finally, we will use these results to prove the existence and uniqueness of lifts that behave well with respect to normal subgroups.
Numerical Simulation of Thermal Sensitivity for Explosives
In order to compare the thermal sensitivity of several typical explosives, this article solved numerically one-dimensional transient heat conduction equation including thermal decomposition for explosives with finite difference method. The explicit difference equations under different boundary conditions were established, and the corresponding stability criterion was derived. The ignition times for typical explosives were calculated, which was used to evaluate thermal sensitivity of explosives.