A subgroup H of a finite G is said to be c-normal in G if there exists a normal subgroup N of G such that G = HN with H ∩ N ≤ HG = CoreG(H). We are interested in studying the influence of the c–normality of certain subgroups of prime power order on the structure of finite groups.

We extend the normal surface Q-theory to non-compact 3-manifolds with respect to ideal triangulations. An ideal triangulation of a 3-manifold often has a small number of tetrahedra resulting in a system of Q-matching equations with a small number of variables. A unique feature of our approach is that a compact surface F with boundary properly embedded in a non-compact 3-manifold M with an ideal triangulation with torus cusps can be represented by a normal surface in M as follows. A half-open annulus made up of an infinite number of triangular disks is attached to each boundary component of F. The resulting surface , when normalized, will contain only a finite number of Q-disks and thus correspond to an admissible solution to the system of Q-matching equations. The correspondence is bijective.

We develop a notion of a *-product on a general abelian group, establish a Weyl calculus for operators on the group and connect these with the representation theory of an associated Heisenberg group. This can all be viewed as a generalization of the familiar theory for R. A symplectic group is introduced and a connection with the classical Cayley transform is established. Our main application is to finite groups, where consideration of the symbol calculus for the cyclic groups provides an interesting alternative to the usual matrix form for linear transformations. This leads to a new basis for sl(n) and a decomposition of this Lie algebra into a sum of C*artan subalgebras.

In this paper we derive precise tail-area approximations for the sum of an arbitrary finite number of independent heavy-tailed random variables. In order to achieve second-order asymptotics, a mild regularity condition is imposed on the class of distribution functions with regularly varying tails.

Higher-order asymptotics are also obtained when considering asemiparametric subclass of distribution functions with regularly varying tails. These semiparametric subclasses are shown to be closed under convolutions and a convolution algebra is constructed to evaluate the parameters of a convolution from the parameters of the constituent distributions in the convolution. A Maple code is presented which does this task.

A geometric mass concerning supersingular abelian varieties with real multiplications is formulated and related to an arithmetic mass. We determine the exact geometric mass formula for superspecial abelian varieties of Hubert-Blumenthal type. As an application, we compute the number of the irreducible components of the supersingular locus of some Hubert-Blumenthal varieties in terms of special values of the zeta function.

This paper gives a characterisation of finite supersoluble groups of Wielandt length two of order coprime to six.

Let L be a finite-dimensional Lie algebra over the field F. The Ado-Iwasawa Theorem asserts the existence of a finite-dimensional L-module which gives a faithful representation ρ of L. Let S be a subnormal subalgebra of L, let be a saturated formation of soluble Lie algebras and suppose that S ∈ . I show that there exists a module V with the extra property that it is -hypercentral as S-module. Further, there exists a module V which has this extra property simultaneously for every such S and , along with the Hochschild extra that ρ(x) is nilpotent for every x ∈ L with ad(x) nilpotent. In particular, if L is supersoluble, then it has a faithful representation by upper triangular matrices.

Let G be a compact abelian group and 1< p < ∞. It is known that the spectrum σ (Tψ) of a Fourier p–multiplier operator Tψ acting in Lp(G), may fail to coincide with its natural spectrum ψ(Г) if p ≠ 2; here Γ is the dual group to G and the bar denotes closure in C. Criteria are presented, based on geometric, topological and/or algebraic properties of the compact set σ(Tψ), which are sufficient to ensure that the equality σ(Tψ) = ψ(Г)holds.

A subgroup H of a finite group G is said to be c–supplemented in G if there exists a subgroup K of G such that G = HK and H∩K is contained in coreG (H). In this paper some results for finite p–nilpotent groups are given based on some subgroups of Pc–supplemented in G, where p is a prime factor of the order of G and P is a Sylow p–subgroup of G. We also give some applications of these results.