Small, $nm$-stable compact $G$-groups

Krzysztof Krupinski (1) , Frank Olaf Wagner (2)

1 Instytut Matematyczny
2 ICJ - Institut Camille Jordan

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.

Keywords: Polish structure; Profinite group; Small compact $G$-group

Domains: Logic [math.LO]; Group Theory [math.GR]

Krzysztof Krupinski, Frank Olaf Wagner. Small, $nm$-stable compact $G$-groups. Israel Journal of Mathematics, 2013, 194 (2), pp.907-933. ⟨10.1007/s11856-012-0103-3⟩. ⟨hal-00495141v4⟩