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101. 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

102. CMB 2001 (vol 44 pp. 27)

Goodaire, Edgar G.; Milies, César Polcino
Normal Subloops in the Integral Loop Ring of an $\RA$ Loop
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.

Categories:20N05, 17D05, 16S34, 16U60

103. 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 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.

Keywords:two-complex, covering space, Cockcroft two-complex, Thompson's group
Categories:57M20, 20F38, 57M10, 20F34

104. CMB 2000 (vol 43 pp. 79)

König, Steffen
Cyclotomic Schur Algebras and Blocks of Cyclic Defect
An explicit classification is given of blocks of cyclic defect of cyclotomic Schur algebras and of cyclotomic Hecke algebras, over discrete valuation rings.

Categories:20G05, 20C20, 16G30, 17B37, 57M25

105. CMB 1999 (vol 42 pp. 335)

Kim, Goansu; Tang, C. Y.
Cyclic Subgroup Separability of HNN-Extensions with Cyclic Associated Subgroups
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.

Keywords:HNN-extension, nilpotent groups, cyclic subgroup separable $(\pi_c)$, residually finite
Categories:20E26, 20E06, 20F10

106. CMB 1999 (vol 42 pp. 298)

Jespers, Eric; Okniński, Jan
Semigroup Algebras and Maximal Orders
We describe contracted semigroup algebras of Malcev nilpotent semigroups that are prime Noetherian maximal orders.

Categories:16S36, 16H05, 20M25

107. 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.


108. 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 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.

Categories:20G10, 11F75

109. CMB 1998 (vol 41 pp. 423)

Long, D. D.; Reid, A. W.
Free products with amalgamation and $\lowercase{p}$-adic Lie groups
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.


110. 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 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

111. 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 $.


112. CMB 1998 (vol 41 pp. 109)

Tahara, Ken-Ichi; 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

113. 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 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

114. CMB 1997 (vol 40 pp. 330)

Kapovich, Ilya
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

115. 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 $n-m$ generators, then $G$ is free of rank $n-m$.

Categories:20E05, 20C99, 14Q99

116. CMB 1997 (vol 40 pp. 266)

Bechtell, H.; Deaconescu, M.; Silberberg, Gh.
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

117. CMB 1997 (vol 40 pp. 341)

Lee, Hyang-Sook
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

118. 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
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