Let R be a ring with identity and let Ω be a totally ordered set. Let Ω′ be a totally ordered set which is disjoint from and equipotent to Ω′ with ′:Ω→Ω′ an order preserving bijection. Define Ω1=Ω∪Ω′ and let Ω1, be totally ordered by inheriting the order from Ω and Ω′ and with ω<λ for all ω∈Ω and λ′∈Ω′. Let M be the free R-module R(Ω1).(We define the alternate bilinear form (*, *) on M by

We are concerned here with question: to what extent can the structure of a group G be recaptured from information about the structure of its group of automorphismsAut G? For example, one might try to find all groups which have some specific group astheir (full) automorphism group, a point of view adopted by Iyer in a recent paper [5]. Nothing is known about this question in general except the result of Nagrebeckü [7] that there are only finitely many finite groups with a given group as automorphismgroup.

There are a number of classes of distributive lattices whose members can be characterised as the coproduct A * L of suitable distributive lattices A and L. For example, Post algebras [1], pseudo-Post algebras [4], Post Lalgebras ([6], [8[9]) and the lattices [D]n of [4]. Moreover, the α-completeness and α-representability of some(though not all) of these algebras have been investigated in [7], [2], [6], and [10].

If F is a (commutative) field let denote the class of all groups G such that every irreducible FG-module has finite dimension over F. The introduction to [7] contains motivation for considering these classes and surveys some of the results to date concerning them. In [7] for every field F we determined the finitely generated soluble groups in . Here, for fields F of characteristic zero, we determine, at least in principle, the soluble groups in . Our main result is the following.

In this paper we investigate the weighted convolution algebras lp(ωn), where 1≦p<∞ and {ωn} is a sequence of positive weights satisfying the following conditions. If p = 1 we require ω0 = 1, and ωt+s≦ωtωs.Then

In studying the cohomology of the symmetric groups and its applications in topology one is led to certain questions concerning the representation rings of special subgroups of . In this note we calculate the Chern classes of the regular representation of (Z/p)n where p is a fixed odd prime in terms of certain modular invariants first described by L. E. Dickson in 1911. In a later paper [9] we apply these results to study the odd primary torsion in the PL cobordism ring. Some indications of this application are given in Sections 10–12 where we apply the result above to obtain information about the cohomology of . After circulation of this note in preprint form we learned that H. Mui [10], has also proved Theorem 6.2.

Let ‖x‖ denote the distance of x from the nearest integer. In 1948 H. Heilbronn proved [5] that for ε>0 and N>c1(ε) the inequality

holds for any real α. This result has since been generalised in many different directions, and we consider here extensions of the type: For ε>0, N>c2{ε, s) and a quadratic formQ(x1,…, xs) there exist integersn1,…,nsnot all zero with |n1|,…,|ns≦N and with