Generalizing and strengthening some well-known
results of Higman, B. Neumann, Hanna Neumann and
Dark on embeddings into two-generator groups, we
introduce a construction of subnormal verbal
embedding of an arbitrary (soluble, fully
ordered or torsion free) ordered countable group
into a two-generator ordered group with these
properties. Further, we establish subnormal
verbal embedding of defect two of an arbitrary
(soluble, fully ordered or torsion free) ordered
group G
into a group with these properties and of the
same cardinality as G,
and show in connection with a problem of
Heineken that the defect of such an embedding
cannot be made smaller, that is, such verbal
embeddings of ordered groups cannot in general be
normal.
|