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A reconstruction theorem for locally moving groups acting on completely metrizable spaces

Edmund Ben-Ami (2010)

Fundamenta Mathematicae

Let G be a group which acts by homeomorphisms on a metric space X. We say the action of G is locally moving on X if for every open U ⊆ X there is a g ∈ G such that g↾X ≠ Id while g↾(X∖U) = Id. We prove the following theorem: Theorem A. Let X,Y be completely metrizable spaces and let G be a group which acts on X and Y with locally moving actions. If the orbits of the action of G on X are of the second category in X and the orbits of the action of G on Y are of the second category...

A relatively free topological group that is not varietal free

Vladimir Pestov, Dmitri Shakhmatov (1998)

Colloquium Mathematicae

Answering a 1982 question of Sidney A. Morris, we construct a topological group G and a subspace X such that (i) G is algebraically free over X, (ii) G is relatively free over X, that is, every continuous mapping from X to G extends to a unique continuous endomorphism of G, and (iii) G is not a varietal free topological group on X in any variety of topological groups.

A remark on a Theorem of J. G. Thompson

Bertram Huppert (1998)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

An important theorem by J. G. Thompson says that a finite group G is p -nilpotent if the prime p divides all degrees (larger than 1) of irreducible characters of G . Unlike many other cases, this theorem does not allow a similar statement for conjugacy classes. For we construct solvable groups of arbitrary p -lenght, in which the lenght of any conjugacy class of non central elements is divisible by p .

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