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Register a new userOn thin-complete ideals of subsets of groups
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Given a family $F$ of subsets of a group $G$ we describe the structure of its thin-completion $tau^*(F)$, which is the smallest thin-complete family that contains $I$. A family $F$ of subsets of $G$ is called thin-complete if each $F$-thin subset of $G$ belongs to $F$. A subset $A$ of $G$ is called $F$-thin if for any distinct points $x,y$ of $G$ the intersection $xAcap yA$ belongs to the family $F$. We prove that the thin-completion of an ideal in an ideal. If $G$ is a countable non-torsion group, then the thin-completion $tau^*(F_G)$ of the ideal $F_G$ of finite subsets of $G$ is coanalytic but not Borel in the power-set $P_G$ of $G$.
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A subset $D$ of an Abelian group is $decomposable$ if $emptyset e Dsubset D+D$. In the paper we give partial answer to an open problem asking whether every finite decomposable subset $D$ of an Abelian group contains a non-empty subset $Zsubset D$ with $sum Z=0$. For every $ninmathbb N$ we present a decomposable subset $D$ of cardinality $|D|=n$ in the cyclic group of order $2^n-1$ such that $sum D=0$, but $sum T e 0$ for any proper non-empty subset $Tsubset D$. On the other hand, we prove that every decomposable subset $Dsubsetmathbb R$ of cardinality $|D|le 7$ contains a non-empty subset $Zsubset D$ of cardinality $|Z|lefrac12|D|$ with $sum Z=0$. For every $ninmathbb N$ we present a subset $Dsubsetmathbb Z$ of cardinality $|D|=2n$ such that $sum Z=0$ for some subset $Zsubset D$ of cardinality $|Z|=n$ and $sum T e 0$ for any non-empty subset $Tsubset D$ of cardinality $|T|<n=frac12|D|$. Also we prove that every finite decomposable subset $D$ of an Abelian group contains two non-empty subsets $A,B$ such that $sum A+sum B=0$.
In this paper we study the generic, i.e., typical, behavior of finitely generated subgroups of hyperbolic groups and also the generic behavior of the word problem for amenable groups. We show that a random set of elements of a nonelementary word hyperbolic group is very likely to be a set of free generators for a nicely embedded free subgroup. We also exhibit some finitely presented amenable groups for which the restriction of the word problem is unsolvable on every sufficiently large subset of words.
A finite group $G$ is called a Schur group, if any Schur ring over $G$ is associated in a natural way with a subgroup of $Sym(G)$ that contains all right translations. Recently, the authors have completely identified the cyclic Schur groups. In this paper it is shown that any abelian Schur group belongs to one of several explicitly given families only. In particular, any non-cyclic abelian Schur group of odd order is isomorphic to $Z_3times Z_{3^k}$ or $Z_3times Z_3times Z_p$ where $kge 1$ and $p$ is a prime. In addition, we prove that $Z_2times Z_2times Z_p$ is a Schur group for every prime $p$.
We study group algebras for compact groups in the category of real and complex weakly complete vector spaces. We also show that the group algebra is a quotient of the weakly complete universal enveloping algebra of the Lie algebra of the compact group. We relate this to Tannaka duality, and to functorial properties of the group algebra. We determine the structure of the group algebra in terms of the irreducible representation, both in the real and the complex case. The particular case of a compact abelian group is worked out in detail.
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