We explicitly describe, in terms of indecomposable $\mathbb{Z}_2 [G]$-modules, the Galois module structure of ideals in totally ramified biquadratic extensions of local number fields with only one break in their ramification filtration. This paper completes work begun in [Elder: Canad. J.~Math. (5) {\bf 50}(1998), 1007--1047].

Let $F$ be a totally real number field and let $\GL_{n}$ be the general linear group of rank $n$ over $F$. Let $\mathfrak{p}$ be a prime ideal of $F$ and $F_{\mathfrak{p}}$ the completion of $F$ with respect to the valuation induced by $\mathfrak{p}$. We will consider a finite quotient of the affine building of the group $\GL_{n}$ over the field $F_{\mathfrak{p}}$. We will view this object as a hypergraph and find a set of commuting operators whose sum will be the usual adjacency operator of the graph underlying the hypergraph.

We show that an $\RA$ loop has a torsion-free normal complement in the loop of normalized units of its integral loop ring. We also investigate whether an $\RA$ loop can be normal in its unit loop. Over fields, this can never happen.

A challenge by R.~Padmanabhan to prove by group theory the commutativity of cancellative semigroups satisfying a particular law has led to the proof of more general semigroup laws being equivalent to quite simple ones.

In a study of the word problem for groups, R.~J.~Thompson considered a certain group $F$ of self-homeomorphisms of the Cantor set and showed, among other things, that $F$ is finitely presented. Using results of K.~S.~Brown and R.~Geoghegan, M.~N.~Dyer showed that $F$ is the fundamental group of a finite two-complex $Z^2$ having Euler characteristic one and which is {\em Cockcroft}, in the sense that each map of the two-sphere into $Z^2$ is homologically trivial. We show that no proper covering complex of $Z^2$ is Cockcroft. A general result on Cockcroft properties implies that no proper regular covering complex of any finite two-complex with fundamental group $F$ is Cockcroft.

An explicit classification is given of blocks of cyclic defect of cyclotomic Schur algebras and of cyclotomic Hecke algebras, over discrete valuation rings.

We derive a necessary and sufficient condition for HNN-extensions of cyclic subgroup separable groups with cyclic associated subgroups to be cyclic subgroup separable. Applying this, we explicitly characterize the residual finiteness and the cyclic subgroup separability of HNN-extensions of abelian groups with cyclic associated subgroups. We also consider these residual properties of HNN-extensions of nilpotent groups with cyclic associated subgroups.

We describe contracted semigroup algebras of Malcev nilpotent semigroups that are prime Noetherian maximal orders.

In this note we further our investigation of Baer invariants of groups by obtaining, as consequences of an exact sequence of A.~S.-T.~Lue, some numerical inequalities for their orders, exponents, and generating sets. An interesting group theoretic corollary is an explicit bound for $|\gamma_{c+1}(G)|$ given that $G/Z_c(G)$ is a finite $p$-group with prescribed order and number of generators.

We study metaplectic coverings of the adelized group of a split connected reductive group $G$ over a number field $F$. Assume its derived group $G'$ is a simply connected simple Chevalley group. The purpose is to provide some naturally defined sections for the coverings with good properties which might be helpful when we carry some explicit calculations in the theory of automorphic forms on metaplectic groups. Specifically, we \begin{enumerate} \item construct metaplectic coverings of $G({\Bbb A})$ from those of $G'({\Bbb A})$; \item for any non-archimedean place $v$, show the section for a covering of $G(F_{v})$ constructed from a Steinberg section is an isomorphism, both algebraically and topologically in an open subgroup of $G(F_{v})$; \item define a global section which is a product of local sections on a maximal torus, a unipotent subgroup and a set of representatives for the Weyl group.

Using the theory of $p$-adic Lie groups we give conditions for a finitely generated group to admit a splitting as a non-trivial free product with amalgamation. This can be viewed as an extension of a theorem of Bass.

The growth series of compact hyperbolic Coxeter groups with 4 and 5 generators are explicitly calculated. The assertions of J.~Cannon and Ph.~Wagreich for the 4-generated groups, that the poles of the growth series lie on the unit circle, with the exception of a single real reciprocal pair of poles, are verified. We also verify that for the 5-generated groups, this phenomenon fails.

We investigate the problem of determining when $\IA (F_{n}({\bf A}_{m}{\bf A}))$ is finitely generated for all $n$ and $m$, with $n\geq 2$ and $m\neq 1$. If $m$ is a nonsquare free integer then $\IA(F_{n}({\bf A}_{m}{\bf A}))$ is not finitely generated for all $n$ and if $m$ is a square free integer then $\IA(F_{n}({\bf A}_{m}{\bf A}))$ is finitely generated for all $n$, with $n\neq 3$, and $\IA(F_{3}({\bf A}_{m}{\bf A}))$ is not finitely generated. In case $m$ is square free, Bachmuth and Mochizuki claimed in ([7], Problem 4) that $\TR({\bf A}_{m}{\bf A})$ is $1$ or $4$. We correct their assertion by proving that $\TR({\bf A}_{m}{\bf A})=\infty $.

Let $G$ be any group, and $H$ be a normal subgroup of $G$. Then M.~Hartl identified the subgroup $G \cap(1+\triangle^3(G)+\triangle(G)\triangle(H))$ of $G$. In this note we give an independent proof of the result of Hartl, and we identify two subgroups $G\cap(1+\triangle(H)\triangle(G)\triangle(H)+\triangle([H,G])\triangle(H))$, $G\cap(1+\triangle^2(G)\triangle(H)+\triangle(K)\triangle(H))$ of $G$ for some subgroup $K$ of $G$ containing $[H,G]$.

In this paper we prove the following: 1.~~Let $m\ge 2$, $n\ge 1$ be integers and let $G$ be a group such that $(XY)^n = (YX)^n$ for all subsets $X,Y$ of size $m$ in $G$. Then \item{a)} $G$ is abelian or a $\BFC$-group of finite exponent bounded by a function of $m$ and $n$. \item{b)} If $m\ge n$ then $G$ is abelian or $|G|$ is bounded by a function of $m$ and $n$. 2.~~The only non-abelian group $G$ such that $(XY)^2 = (YX)^2$ for all subsets $X,Y$ of size $2$ in $G$ is the quaternion group of order $8$. 3.~~Let $m$, $n$ be positive integers and $G$ a group such that $$ X_1\cdots X_n\subseteq \bigcup_{\sigma \in S_n\bs 1} X_{\sigma (1)} \cdots X_{\sigma (n)} $$ for all subsets $X_i$ of size $m$ in $G$. Then $G$ is $n$-permutable or $|G|$ is bounded by a function of $m$ and $n$.

This note contains the classification of the finite groups $G$ satisfying the condition $N_{G}(H)/C_{G}(H)\cong \Aut(H)$ for every abelian subgroup $H$ of $G$.

Using naive algebraic geometric methods a new proof of the following celebrated theorem of Magnus is given: Let $G$ be a group with a presentation having $n$ generators and $m$ relations. If $G$ also has a presentation on $n-m$ generators, then $G$ is free of rank $n-m$.

The main purpose of this paper is to study groups $G_1$, $G_2$ such that $H^\ast(BG_1,{\bf Z}/p)$ is isomorphic to $H^\ast(BG_2,{\bf Z}/p)$ in ${\cal U}$, the category of unstable modules over the Steenrod algebra ${\cal A}$, but not isomorphic as graded algebras over ${\bf Z}/p$.

We show that if $A$ is a torsion-free word hyperbolic group which belongs to class $(Q)$, that is all finitely generated subgroups of $A$ are quasiconvex in $A$, then any maximal cyclic subgroup $U$ of $A$ is a Burns subgroup of $A$. This, in particular, implies that if $B$ is a Howson group (that is the intersection of any two finitely generated subgroups is finitely generated) then $A\ast_U B$, $\langle A,t \mid U^t=V\rangle$ are also Howson groups. Finitely generated free groups, fundamental groups of closed hyperbolic surfaces and some interesting $3$-manifold groups are known to belong to class $(Q)$ and our theorem applies to them. We also describe a large class of word hyperbolic groups which are not Howson.

Dimension subgroups and Lie dimension subgroups are known to satisfy a `universal coefficient decomposition', {\it i.e.} their value with respect to an arbitrary coefficient ring can be described in terms of their values with respect to the `universal' coefficient rings given by the cyclic groups of infinite and prime power order. Here this fact is generalized to much more general types of induced subgroups, notably covering Fox subgroups and relative dimension subgroups with respect to group algebra filtrations induced by arbitrary $N$-series, as well as certain common generalisations of these which occur in the study of the former. This result relies on an extension of the principal universal coefficient decomposition theorem on polynomial ideals (due to Passi, Parmenter and Seghal), to all additive subgroups of group rings. This is possible by using homological instead of ring theoretical methods.