Algebra in the superextensions of twinic groups
Tom 473 / 2010
Streszczenie
\def\mathsf#1{\hbox{\font\sf=cmss10\sf#1}}% \def\Enl{{\rm End}_\lambda}\def\I{{\cal I}}% \def\IK{\mathsf K}\def\IZ{{\sym Z}}\def\IN{{\sym N}}% \def\w{\omega}% Given a group $X$ we study the algebraic structure of the compact right-topological semigroup $\lambda(X)$ consisting of all maximal linked systems on $X$. This semigroup contains the semigroup $\beta(X)$ of ultrafilters as a closed subsemigroup. We construct a faithful representation of the semigroup $\lambda(X)$ in the semigroup $\mathsf P(X)^{\mathsf P(X)}$ of all self-maps of the power-set $\mathsf P(X)$ and show that the image of $\lambda(X)$ in $\mathsf P(X)^{\mathsf P(X)}$ coincides with the semigroup $\Enl(\mathsf P(X))$ of all functions $f:\mathsf P(X)\to\mathsf P(X)$ that are equivariant, monotone and symmetric in the sense that $f(X\setminus A)=X\setminus f(A)$ for all $A\subset X$. Using this representation we describe the minimal ideal $\IK(\lambda(X))$ and minimal left ideals of the superextension $\lambda(X)$ of a twinic group $X$. A group $X$ is called {\it twinic} if it admits a left-invariant ideal $\I\subset\mathsf P(X)$ such that $xA=_\I yA$ for all subsets $A\subset X$ and points $x,y\in X$ with $xA\subset_\I X\setminus A\subset_\I yA$. The class of twinic groups includes all amenable groups and all groups with periodic commutators but does not include the free group $F_2$ with two generators. We prove that for any twinic group $X$, there is a cardinal $m$ such that all minimal left ideals of $\lambda(X)$ are algebraically isomorphic to $$2^m\times \prod_{1\le k\le\infty}C_{2^k}^{q(X,C_{2^k})}\times \prod_{3\le k\le\infty}Q_{2^k}^{q(X,C_{2^k})}$$ for some cardinals $q(X,C_{2^k})$ and $q(X,Q_{2^k})$, $k\in\IN\cup\{\infty\}$. Here $C_{2^k}$ is the cyclic group of order $2^k$, $C_{2^\infty}$ is the quasicyclic 2-group and $Q_{2^k}$, $k\in\IN\cup\{\infty\}$, are the groups of generalized quaternions. If the group $X$ is abelian, then $q(X,Q_{2^k})=0$ for all $k$ and $q(X,C_{2^k})$ is the number of subgroups $H\subset X$ with quotient $X/H$ isomorphic to $C_{2^k}$. If $X$ is an abelian group (admitting no epimorphism onto $C_{2^\infty}$), then each minimal left ideal of the superextension $\lambda(X)$ is algebraically (and topologically) isomorphic to the product $\prod_{1\le k\le\infty} (C_{2^k}\times 2^{2^{k-1}-k})^{q(X,C_{2^k})}$ where the cube $2^{2^{k-1}-k}$ (equal to $2^\w$ if $k=\infty$) is endowed with the left zero multiplication. For an abelian group $X$, all minimal left ideals of $\lambda(X)$ are metrizable if and only if $X$ has finite ranks $r_0(X)$ and $r_2(X)$ and admits no homomorphism onto the group $C_{2^\infty}\oplus C_{2^\infty}$. Applying this result to the group $\IZ$ of integers, we prove that each minimal left ideal of $\lambda(\IZ)$ is topologically isomorphic to $2^\w\times\prod_{k=1}^\infty C_{2^k}$. Consequently, all subgroups in the minimal ideal $\IK(\lambda(\IZ))$ of $\lambda(\IZ)$ are profinite abelian groups. On the other hand, the superextension $\lambda(\IZ)$ contains an isomorphic topological copy of each second countable profinite topological semigroup. This results contrasts with the famous Zelenyuk Theorem saying that the semigroup $\beta(\IZ)$ contains no finite subgroups. At the end of the paper we describe the structure of minimal left ideals of finite groups $X$ of order $|X|\le 15$.
