101. CMB 2001 (vol 44 pp. 385)
 Ballantine, Cristina M.

A Hypergraph with Commuting Partial Laplacians
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.
Keywords:Hecke operators, buildings Categories:11F25, 20F32 

102. CMB 2001 (vol 44 pp. 93)
 Neumann, B. H.

Some Semigroup Laws in Groups
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.
Categories:20E10, 20M07 

103. CMB 2001 (vol 44 pp. 27)
104. CMB 2000 (vol 43 pp. 268)
 Bogley, W. A.; Gilbert, N. D.; Howie, James

Cockcroft Properties of Thompson's Group
In a study of the word problem for groups, R.~J.~Thompson
considered a certain group $F$ of selfhomeomorphisms 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 twocomplex $Z^2$
having Euler characteristic one and which is {\em Cockcroft}, in
the sense that each map of the twosphere 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
twocomplex with fundamental group $F$ is Cockcroft.
Keywords:twocomplex, covering space, Cockcroft twocomplex, Thompson's group Categories:57M20, 20F38, 57M10, 20F34 

105. CMB 2000 (vol 43 pp. 79)
106. CMB 1999 (vol 42 pp. 335)
 Kim, Goansu; Tang, C. Y.

Cyclic Subgroup Separability of HNNExtensions with Cyclic Associated Subgroups
We derive a necessary and sufficient condition for HNNextensions
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 HNNextensions of abelian groups with
cyclic associated subgroups. We also consider these residual
properties of HNNextensions of nilpotent groups with cyclic
associated subgroups.
Keywords:HNNextension, nilpotent groups, cyclic subgroup separable $(\pi_c)$, residually finite Categories:20E26, 20E06, 20F10 

107. CMB 1999 (vol 42 pp. 298)
108. CMB 1998 (vol 41 pp. 385)
 Burns, John; Ellis, Graham

Inequalities for Baer invariants of finite groups
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.
Category:20C25 

109. CMB 1998 (vol 41 pp. 488)
 Sun, Heng

Remarks on certain metaplectic groups
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 nonarchimedean 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.
Categories:20G10, 11F75 

110. CMB 1998 (vol 41 pp. 423)
111. CMB 1998 (vol 41 pp. 231)
 Worthington, R. L.

The growth series of compact hyperbolic Coxeter groups with 4 and 5 generators
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 4generated 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 5generated groups, this
phenomenon fails.
Categories:20F05, 20F55 

112. CMB 1998 (vol 41 pp. 98)
 Papistas, Athanassios I.

Automorphisms of metabelian groups
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 $.
Category:20F28 

113. CMB 1998 (vol 41 pp. 109)
 Tahara, KenIchi; Vermani, L. R.; Razdan, Atul

On generalized third dimension subgroups
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 $[H,G]$.
Categories:20C07, 16S34 

114. CMB 1998 (vol 41 pp. 65)
 Mohammadi Hassanabadi, A.; Rhemtulla, Akbar

Criteria for commutativity in large groups
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 nonabelian 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 $n$.
Categories:20E34, 20F24 

115. CMB 1997 (vol 40 pp. 330)
 Kapovich, Ilya

Amalgamated products and the Howson property
We show that if $A$ is a torsionfree 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.
Categories:20E06, 20E07, 20F32 

116. CMB 1997 (vol 40 pp. 352)
 Liriano, Sal

A New Proof of a Theorem of Magnus
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 $nm$ generators, then
$G$ is free of rank $nm$.
Categories:20E05, 20C99, 14Q99 

117. CMB 1997 (vol 40 pp. 266)
118. CMB 1997 (vol 40 pp. 341)
 Lee, HyangSook

The stable and unstable types of classifying spaces
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 ${\bf Z}/p$.
Categories:55R35, 20J06 

119. CMB 1997 (vol 40 pp. 47)
 Hartl, Manfred

A universal coefficient decomposition for subgroups induced by submodules of group algebras
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.
Keywords:induced subgroups, group algebras, Fox subgroups, relative dimension, subgroups, polynomial ideals Categories:20C07, 16A27 
