Alfred W. Hales

Nonnegative Hall Polynomials

AbstractThe number of subgroups of type μ and cotype ν in a finite abelian p-group of type λ is a polynomialg

Generalized flags in p-groups

_{\mu v}^\lambda (p)

Jordan decomposition and hypercentral units in integral group rings

with integral coefficients. We prove g

Partial augmentations and jordan decomposition in group ring

_{\mu v}^\lambda (p)

The multiplicative Jordan decomposition in group rings, II

has nonnegative coefficients for all partitions μ and ν if and only if no two parts of λ differ by more than one. Necessity follows from a few simple facts about Hall-Littlewood symmetric functions; sufficiency relies on properties of certain order-preserving surjections ϕ that associate to each subgroup a vector dominated componentwise by λ. The nonzero components of ϕ(H) are the parts of μ, the type of H; if no two parts of λ differ by more than one, the nonzero components of λ − ϕ(H) are the parts of ν, the cotype of H. In fact, we provide an order-theoretic characterization of those isomorphism types of finite abelian p-groups all of whose Hall polynomials have nonnegative coefficients.

The second augmentation quotient of an integral group ring

Abstract Let G be a finite group acting on a finite set S . Burnsides formula expresses the number of orbits of S under G in terms of the numbers I(g) , where I(g) is the number of fixed points of S under the group element g . This paper characterizes the circumstances under which two group elements, g 1 and g 2 , satisfy I(g 1 )= I(g 2 ) for all actions of G . Several related questions are also discussed.