Submission date: 29 June 2010.

Abstract:

We prove that if (H,G) is a small, nm-stable compact G-group, then H is nilpotent-by-finite, and if additionally \NM(H) \leq \omega, then H is abelian-by-finite. Both results are significant steps towards the proof of the conjecture that each small, nm-stable compact G-group is abelian-by-finite. We give examples of small, nm-stable compact G-groups of infinite ordinal \NM-rank, providing counter-examples to the \NM-gap conjecture.

Mathematics Subject Classification: Polish structure - Profinite group - Small compact G-group.

Keywords and phrases:

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