Download e-book for kindle: Compact Transformation Groups, part 2 by Ku H.T. (ed.), Mann I.N., Sicks J.L.

By Ku H.T. (ed.), Mann I.N., Sicks J.L.

Lawsuits Of the second one convention On Compact Tranformation teams. collage Of Massachusetts, Amherst, 1971

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Extra resources for Compact Transformation Groups, part 2

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1, a profinite group cannot be the union of countably many closed subsets with empty interior. Hence Prt (X) contains a nonempty open set U for some t = 1, 2, . . Clearly G = g∈G gU . By r compactness there exist finitely many g1 , . . , gr ∈ G such that G = i=1 gi U . Since G = X , there exists some s such that g1 , . . , gr ∈ P rs (X). Put m = t + s; then G = Prm (X). 5 Finitely Generated Profinite Groups A closed subgroup K of a profinite group is called characteristic if ϕ(K) = K for all continuous automorphisms ϕ of G.

5 (1) The prototype of a direct limit is a union. If an abelian group A is a union A = i∈I Ai of subgroups Ai , then A is the direct limit of the subgroup generated by the finite unions j∈J Aj , where J ranges over the finite subsets of I. Conversely, if A = lim Ai −→ i∈I is a direct limit of a direct system {Ai , ϕij , I}, and if ϕi : Ai −→ A are the canonical maps, then A= ϕi (Ai ). i∈I (2) Every abelian group A is a direct limit of its finitely generated subgroups. In particular, if A is torsion, it is the direct limit of its finite subgroups.

If this were not the case, there would exist an open normal subgroup U of G with U ∩ T < T . We prove that this leads to a contradiction by exhibiting a continuous section ζ : G/H −→ G/(U ∩ T ) of G/(U ∩ T ) −→ G/H such that (U ∩ T, ζ) of ζ, it suffices to find a continuous section (T, σ). 2 Basic Properties of Pro - C Groups 31 to the projection G/(U ∩ T ) −→ G/T. But G/T = (G/(U ∩ T ))/(T /(U ∩ T )), and T /(U ∩ T ) is a finite group. Thus the existence of ξ follows from the first part of the proof.

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