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126. 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

127. 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

128. 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

129. 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

130. 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

131. 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

132. 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

133. 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.

Category:20E06

134. 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

135. 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

136. 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

137. 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

138. 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
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