Notes on non-archimedean topological groups

We show that the Heisenberg type group H X = ( Z 2 ⊕ V ) ⋋ V ⁎ , with the discrete Boolean group V : = C ( X , Z 2 ) , canonically defined by any Stone space X, is always minimal. That is, H X does not admit any strictly coarser Hausdorff group topology. This leads us to the following result: for every (locally compact) non-archimedean G there exists a (resp., locally compact) non-archimedean minimal group M such that G is a group retract of M. For discrete groups G the latter was proved by S. Dierolf and U. Schwanengel (1979) [6]. We unify some old and new characterization results for non-archimedean groups.