Expand all Collapse all | Results 101 - 109 of 109 |
101. CMB 1998 (vol 41 pp. 231)
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 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.
Categories:20F05, 20F55 |
102. CMB 1998 (vol 41 pp. 98)
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 |
103. CMB 1998 (vol 41 pp. 109)
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 |
104. CMB 1998 (vol 41 pp. 65)
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 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 $n$.
Categories:20E34, 20F24 |
105. CMB 1997 (vol 40 pp. 266)
Finite groups with large automizers for their Abelian subgroups 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 $G$.
Categories:20E34, 20D45 |
106. CMB 1997 (vol 40 pp. 352)
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 $n-m$ generators, then
$G$ is free of rank $n-m$.
Categories:20E05, 20C99, 14Q99 |
107. CMB 1997 (vol 40 pp. 341)
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 |
108. CMB 1997 (vol 40 pp. 330)
Amalgamated products and the Howson property 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.
Categories:20E06, 20E07, 20F32 |
109. CMB 1997 (vol 40 pp. 47)
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 |