Algebra in the superextensions of twinic groups

Taras Banakh ; Volodymyr Gavrylkiv

GDML_Books, (2010), p.

Résumé

Given a group X we study the algebraic structure of the compact right-topological semigroup λ(X) consisting of all maximal linked systems on X. This semigroup contains the semigroup β(X) of ultrafilters as a closed subsemigroup. We construct a faithful representation of the semigroup λ(X) in the semigroup $(X)^{(X)}$ of all self-maps of the power-set (X) and show that the image of λ(X) in $(X)^{(X)}$ coincides with the semigroup $End_{λ}((X))$ of all functions f: (X) → (X) that are equivariant, monotone and symmetric in the sense that f(X∖A)=X∖f(A) for all A ⊂ X. Using this representation we describe the minimal ideal (λ(X)) and minimal left ideals of the superextension λ(X) of a twinic group X. A group X is called twinic if it admits a left-invariant ideal ⊂ (X) such that $xA=_{}yA$ for all subsets A ⊂ X and points x,y ∈ X with $xA ⊂ _{}X∖A ⊂ _{}yA$. The class of twinic groups includes all amenable groups and all groups with periodic commutators but does not include the free group F₂ with two generators. We prove that for any twinic group X, there is a cardinal m such that all minimal left ideals of λ(X) are algebraically isomorphic to $2^{m} × ∏_{1≤k≤∞} C_{2^{k}}^{q(X,C_{2^{k}})} × ∏_{3≤k≤∞} Q_{2^{k}}^{q(X,C_{2^{k}})}$ for some cardinals $q(X,C_{2^{k}})$ and $q(X,Q_{2^{k}})$, k ∈ ℕ ∪ ∞. Here $C_{2^{k}}$ is the cyclic group of order $2^{k}$, $C_{2^{∞}}$ is the quasicyclic 2-group and $Q_{2^{k}}$, k ∈ ℕ ∪ ∞, 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 ⊂ X with quotient X/H isomorphic to $C_{2^{k}}$. If X is an abelian group (admitting no epimorphism onto $C_{2^{∞}}$), then each minimal left ideal of the superextension λ(X) is algebraically (and topologically) isomorphic to the product $∏_{1≤k≤∞} (C_{2^{k}} × 2^{2^{k-1}-k})^{q(X,C_{2^{k}})}$ where the cube $2^{2^{k-1}-k}$ (equal to $2^{ω}$ if k = ∞) is endowed with the left zero multiplication. For an abelian group X, all minimal left ideals of λ(X) are metrizable if and only if X has finite ranks r₀(X) and r₂(X) and admits no homomorphism onto the group $C_{2^{∞}} ⊕ C_{2^{∞}}$. Applying this result to the group ℤ of integers, we prove that each minimal left ideal of λ(ℤ) is topologically isomorphic to $2^{ω}×∏_{k=1}^{∞}C_{2^{k}}$. Consequently, all subgroups in the minimal ideal (λ(ℤ)) of λ(ℤ) are profinite abelian groups. On the other hand, the superextension λ(ℤ) contains an isomorphic topological copy of each second countable profinite topological semigroup. This results contrasts with the famous Zelenyuk Theorem saying that the semigroup β(ℤ) 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| ≤ 15.

Détail

Zbl 1225.22004

EUDML-ID : urn:eudml:doc:285993