Sequentially complete groups: dimension and minimality

A topological group G is called sequentially complete if it is sequentially closed in any other topological group (or equivalently, G is sequentially closed in its Raimagekov completion image). We establish the following compactness criterion in the class of connected Abelian groups of non-measurable size: a group in this class is compact iff it is minimal and sequentially complete. We also describe the structure of sequentially complete minimal Abelian groups in the general case. Coincidence of hereditary disconnectedness and zero dimensionality is established for various classes of sequentially complete groups.