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The abelian subgroup conjecture: A counter example. (English) Zbl 0991.22006
A well-known theorem of E. I. Zelmanov [Isr. J. Math. 77, 83-95 (1992; Zbl 0786.22008)] proves that every compact group contains some infinite Abelian subgroup. The results obtained by K. H. Hofmann and S. A. Morris in [Math. Proc. Cambridge Philos. Soc., to appear] suggested the conjecture that every infinite compact group $$G$$ should contain an Abelian subgroup whose topological weight equals that of $$G$$, this is the Abelian subgroup conjecture.
The paper under review disproves this conjecture by showing that the free pro-$$p$$ group $$F_p(X)$$ on any uncountable set $$X$$ constitutes a counterexample: $$F_p(X)$$ has weight $$|X|$$ while all its Abelian subgroups are topologically isomorphic to the additive group of $$p$$-adic integers and therefore have countable weight.
MSC:
 22C05 Compact groups
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