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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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