The Abelian Subgroup Conjecture: A Counter Example

W. Herfort

Abstract: If an abelian subgroup $A$ of a locally compact group $G$ has the same weigth as $G$, it is termed large (see Hofmann, K. H., and S. A. Morris, Compact groups with large abelian subgroups, Math. Proc. Cambridge Philos. Soc. 133 (2002), to appear). It has been conjectured that every compact group has a large abelian subgroup. In this note we show that no free pro-$p$ group $F(X)$ on set $X$ of cardinality greater than $\aleph_0$ contains a large abelian subgroup.

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