Cancellation of topological groups

Abstract

Let M be a family of Hausdorff topological groups. A Hausdorff topological group P is called M-cancellable if for any two topological groups G and H in M, the product group G×P is topologically isomorphic to H×P if and only if G and H are topologically isomorphic. If M is the class of all Hausdorff topological groups, M-cancellable topological groups are called cancellable topological groups.We show in this paper that every topological group in a family N of topological groups is cancellable. In particular, the additive group R of reals and Q of rationals endowed with the usual topologies are both cancellable. We also show that every topological group in a family M of topological groups is cancellable. In particular, every topologically simple non-abelian group is cancellable. It is also proved that every topologically simple Hopfian or co-Hopfian topological group is cancellable.